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+{"text":"\\section{Introduction}\n\n\nPhysics of neutrinos has entered into a new stage \nafter establishment of the mass-induced neutrino oscillation due to \nthe atmospheric~\\cite{atm_evidence}, the accelerator \n~\\cite{K2K_evidence, MINOS}, and \nthe reactor neutrino~\\cite{KL_evidence} experiments, \nconfirming the earlier discovery~\\cite{SKatm,solar,KamLAND} \nand identifying the nature of the phenomenon. \nIn the new era, the experimental endeavors will be focused on search for \nthe unknowns in neutrino masses and the lepton flavor mixing, \n$\\theta_{13}$, the neutrino mass hierarchy, and CP violation. \nOn the theory side, various approaches toward understanding physics of \nlepton mixing and the quark-lepton relations are extensively pursuit \n\\cite{moha-smi}, which then further motivate \nprecision measurement of the lepton mixing parameters. \nWe will use the standard notation~\\cite{PDG} of the lepton mixing matrix, \nthe Maki-Nakagawa-Sakata (MNS) matrix~\\cite{MNS}, \nthroughout this paper.\n\n\nIt was recognized sometime ago that there exists problem of \nparameter degeneracy \nwhich would act as an obstacle against precision measurement of \nthe lepton mixing parameters. \nThe nature of the degeneracy can be understood as \nthe intrinsic degeneracy~\\cite{intrinsic}, \nwhich is duplicated by the unknown sign of atmospheric \n$\\Delta m^2$~\\cite{MNjhep01} (hereafter, ``sign-$\\Delta m^2$ degeneracy'' for \nsimplicity) \nand by the possible octant ambiguity of $\\theta_{23}$~\\cite{octant} \nthat exists if $\\theta_{23}$ is not maximal. \nFor an overview of the resultant eight-fold degeneracy, see e.g., \n\\cite{BMW,MNP2}. \n\n\nIn a previous paper~\\cite{T2KK}, we have shown that \nthe identical two detector setting in Kamioka and in Korea with \neach fiducial mass of 0.27 Mton water, which \nreceives the identical neutrino beam from the J-PARC facility \ncan be sensitive to the neutrino mass hierarchy and CP violation \nin a wide range of the lepton mixing parameters, $\\theta_{13}$ \nand the CP phase $\\delta$.\nIt is the purpose of this paper to point out that the same setting \nhas capability of resolving the $\\theta_{23}$ octant degeneracy \nto a value of $\\theta_{23}$ which is rather close to the maximal, \n$\\sin^2 2 \\theta_{23} < 0.97 (0.94)$ at 2 (3) standard deviation \nconfidence level (CL). \nIt is achieved by detecting solar-$\\Delta m^2$ scale oscillation effect in \nthe Korean detector. \nTogether with the sensitivities to resolution of the degeneracy \nrelated to the mass hierarchy and the CP phase discussed in \nthe previous paper, we demonstrate that \nthe Kamioka-Korea two detector setting is capable of solving \nthe total eight-fold parameter degeneracy.\nWe stress that resolving the degeneracy is crucial to precision \nmeasurement of the lepton mixing parameters on which we make \nfurther comments at appropriate points in the subsequent discussions. \nWe also emphasize that it is highly nontrivial that one can formulate \nsuch a global strategy for resolving all the known degeneracies \n(though in a limited range of the mixing parameters) \nonly with the experimental apparatus using conventional muon neutrino \nsuperbeam.\\footnote{\nIt may be contrasted to the method for resolving the degeneracy \nbased on neutrino factory examined in~\\cite{donini}; \nIt uses a 40 kton magnetized iron calorimeter and a 4 kton emulsion chamber, \nand conventional $\\nu_{\\mu}$ beam watched by \na 400 kton water Cherenkov detector. \n}\n\n\nIn some of the previous analyses including ours~\\cite{T2KK,resolve23}, \npeople often tried to resolve the degeneracy of a particular type \nwithout knowing (or addressing) the solutions of the other types of \ndegeneracies. \nBut, then, the question of consistency of the procedure immediately arises; \nCan one solve the degeneracy of type A without knowing the solutions \nof the other degeneracies B and C? \nDoes the obtained solution remain unchanged when the assumed \nsolutions for the other type of degeneracies are changed to the \nalternative ones?, etc. \nWe answer to these questions in the positive in experimental settings \nwhere the earth matter effect can be treated as perturbation. \nWe do so by showing that the resolution of the degeneracy of a \nparticular type decouples from the remaining degeneracies, \nthe property called the ``decoupling between the degeneracies'' \nin this paper. \n\n\nIn Sec.~\\ref{how}, we present a pedagogical discussion of how the \neight-fold degeneracy can be lifted by measurement with  \nthe Kamioka-Korea two detector setting. \nIn Sec.~\\ref{decoupling}, we prove the ``decoupling'' and make \na brief comment on its significance.\nIn Sec.~\\ref{23degeneracy}, we discuss some characteristic features \nof the $\\nu_{e}$ and $\\bar{\\nu}_{e}$ appearance probabilities \nthat allow the Kamioka-Korea identical two detector setting to \nresolve the $\\theta_{23}$ octant degeneracy. \nIn Sec.~\\ref{sensitivity}, the actual analysis procedure \nand the obtained sensitivities for solving the $\\theta_{23}$ degeneracy \nare described in detail. \nIn Sec.~\\ref{revisit}, we reexamine the sensitivities to \nthe mass hierarchy and CP violating phase by using our new code \nwith disappearance channels and additional systematic errors.\nIn Sec.~\\ref{summary}, we give a summary and discussions. \n\n\n\\section{How the identical two detector system solves the eight-fold degeneracy?}\n\\label{how}\n\n\nWe describe in this section how the eight-fold parameter degeneracy\ncan be resolved by using two identical detectors, one placed at a medium \nbaseline distance of a few times 100 km, and the other at $\\sim$1000 km or so. \nWe denote them as the intermediate and the far detectors, respectively, \nin this paper. \nWhenever necessary we refer the particular setting of Kamioka-Korea \ntwo detector system, but most of the discussions in this and the next \nsections are valid without the specific setting. \n\n\nTo give the readers a level-one understanding we quote here, \nignoring complications, \nwhich effect is most important for solving which degeneracy: \n\n\\begin{itemize}\n\n\\item\n\nThe intrinsic degeneracy; \nSpectrum information solves the intrinsic degeneracy.\n\n\\item\n\nThe sign-$\\Delta m^2$ degeneracy; \nDifference in the earth matter effect between the intermediate and the far detectors \nsolves the sign-$\\Delta m^2$ degeneracy. \n\n\\item\n\nThe $\\theta_{23}$ octant degeneracy; \nDifference in solar $\\Delta m^2$ oscillation effect \n(which is proportional to $c^2_{23}$) between the intermediate \nand the far detectors solves the $\\theta_{23}$ octant degeneracy. \n\n\\end{itemize}\n\n\nTo show how the eight-fold parameter degeneracy can be resolved, \nwe present in Fig~\\ref{intrinsicKamKorea} a comparison between \nthe sensitivities achieved by the Kamioka only setting and the \nKamioka-Korea setting by taking \na particular set of true values of the mixing parameters which are \nquoted in caption of Fig~\\ref{intrinsicKamKorea}. \nThe left four panels of Fig~\\ref{intrinsicKamKorea} show the \nexpected allowed regions of oscillation parameters \nin the Tokai-to-Kamioka phase-II (T2K II) setting, \nwhile the right four panels show the allowed regions by the \nTokai-to-Kamioka-Korea setting.\nFor both settings we assume \n4 years of neutrino plus 4 years of anti-neutrino \nrunning\\footnote{\nIt was shown in the previous study~\\cite{T2KK} that the sensitivity \nobtained with 2 years of neutrino and 6 years of anti-neutrino \nrunning in the T2K II setting~\\cite{T2K} is very similar to that of \n4 years of neutrino and 4 years of anti-neutrino running. \n}\nand the total fiducial volume is kept to be the same, 0.54 Mton.\nSome more information of the experimental setting and the \ndetails of the analysis procedure are described in the caption of \nFig~\\ref{intrinsicKamKorea} and in Sec.~\\ref{sensitivity}. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{sens.kam.1.eps}\n\\includegraphics[width=0.49\\textwidth]{sens.kam_kor.1.eps}\n\\end{center}\n\\caption{The region allowed in $\\delta-\\sin^2 2\\theta_{13}$ and \n$\\sin^2 \\theta_{23}-\\sin^2 2\\theta_{13}$ spaces \nby T2K II (left four panels) and by the \nKamioka-Korea two detector setting (right four panels) in both of which  \n4 years of neutrino plus 4 years of anti-neutrino running are assumed.  \nThe upper (lower) four panels show the allowed region for\nthe positive (negative) sign of $\\Delta m^2_{31}$. \nThe detector fiducial volumes of T2K II and Kamioka-Korea settings \nare assumed to be 0.54 Mton and each 0.27 Mton, respectively, \nand the beam power of J-PARC is assumed to be 4 MW. \nThe baseline to the Kamioka and Korea detectors are, \n295 km and  1050 km, respectively. \nThe true solution is assumed to be located at \n$\\sin^2{2\\theta_{13}}$=0.01, $\\sin ^2 \\theta_{23}$=0.60\nand $\\delta$=$\\pi\/4$  \nwith positive sign of $\\Delta m_{31}^2 (=+2.5\\times 10^{-3}$ eV$^2$), which is \nindicated by the green star. \nThe solar mixing parameters are fixed as\n$\\Delta m_{21}^2 = 8\\times 10^{-5}$ eV$^2$ and $\\sin^2{\\theta_{12}}$=0.31.\nThree contours in each figure correspond to\nthe 68\\% (blue line), 90\\% (black line) and 99\\% \n(red line) C.L. sensitivities, which are defined\nas the difference of the $\\chi^2$ being 2.30, 4.61 and 9.21, respectively.\n}\n\\label{intrinsicKamKorea}\n\\end{figure}\n\n\nLet us first focus on the left four panels of Fig~\\ref{intrinsicKamKorea}. \nIn the left-most two panels labeled as (aN) and (aI), one observes \nsome left-over degeneracies of the total eight-fold degeneracy; \nIf we plot the result of a rate only \nanalysis without spectrum information we would have seen \n8 separate (or overlapped) allowed parameter regions.  \nThe $\\theta_{23}$ octant degeneracy remains unresolved \nas seen in panels (bN) and (bI). \nNote that the overlapping two regions in (aN) and (aI) are \nnothing but the consequence of unresolved $\\theta_{23}$ degeneracy. \nThe intrinsic degeneracy, horizontal pair seen in (aN), is almost \nresolved apart from 99\\% CL region at the particular set of values \nof the mixing parameters indicated above. \nThe corresponding pair in (aI) is missing because the \nintrinsic degeneracy is completely lifted.\nSince the matter effect plays minor role in the T2K II setting it is likely \nthat the spectral information is mainly responsible for lifting the intrinsic \ndegeneracy. See Sec.~\\ref{intrinsic} for more about it. \n\n\nHere is a brief comment on the property of the intrinsic and the \nsign-$\\Delta m^2$ degeneracies. \nBecause the degenerate solutions of CP phase $\\delta$ satisfy \napproximately the same relationship $\\delta_2 = \\pi - \\delta_1$ \nin both the intrinsic and the sign-$\\Delta m^2$ degeneracies \n\\cite{intrinsic,MNjhep01} \n(see Eqs.~(\\ref{intrinsicDS}) and (\\ref{signdm2DS}) in Sec.~\\ref{decoupling}), \nthe would-be four (one missing) regions in the panels (aN) and (aI) in \nFig.~\\ref{intrinsicKamKorea} forms a cross (or X) shape, \nwith crossing connection between a pair of solutions of \nthe sign-$\\Delta m^2$ degeneracy. \n\n\nIn the right four panels of Fig.~\\ref{intrinsicKamKorea} it is exhibited that \nthe intrinsic degeneracy as well as $\\theta_{23}$ octant degeneracy \nare completely resolved by the Kamioka-Korea \ntwo-detector setting at the same values of the mixing parameters, \nindicating power of the two detector method~\\cite{MNplb97}. \nNamely, the comparison between the spectral shapes in Kamioka \nand in Korea located at the first and nearly the second \noscillation maxima, respectively, supersedes a single detector \nmeasurement in Kamioka with the same total volume \ndespite much less statistics in the Korean detector. \nWe will give a detailed discussion on how $\\theta_{23}$ octant \ndegeneracy can be resolved by the Kamioka-Korea setting in \nSec.~\\ref{23degeneracy}, and present the details of the analysis \nin Sec.~\\ref{sensitivity}.\n\n\nIt should be noted that the sign-$\\Delta m^2$ degeneracy is also \nlifted though incompletely at the particular set of values of the \nmixing parameters as indicated in the panels (cI) \nof Fig.~\\ref{intrinsicKamKorea} where only the 99\\% CL regions remain. \nIn fact, we have shown in our previous paper that the \nKamioka-Korea identical two detector setting is powerful in \nresolving the sign-$\\Delta m^2$ degeneracy in a wide range of the \nmixing parameters~\\cite{T2KK}. \nWe note that resolution of the degeneracy in turn leads to an \nenhanced sensitivity to CP violation than that of T2K II setting in \na region of relatively large $\\theta_{13}$. \nSee~\\cite{T2KK} for comparison \nwith T2K II sensitivity. \nAltogether, we verify that the identical two detector setting in \nKamioka-Korea with neutrino beam from J-PARC solves all the \neight-fold parameter degeneracy {\\em in situ} if $\\theta_{13}$ \nis within reach by the next generation superbeam experiments \nsuch as T2K~\\cite{T2K} and NO$\\nu$A~\\cite{NOvA}.\n\n\n\\section{Decoupling between degeneracies}\n\\label{decoupling}\n\n\nIn this section we discuss the property called the \n``decoupling between degeneracies'' which arises due to the \nspecial setting of baselines shorter than $\\sim$1000 km. \nThe content of this section is somewhat independent of the main line \nof the discussion in this paper, and the readers can skip it \nto go directly to the analysis of $\\theta_{23}$ octant degeneracy \nin Secs.~\\ref{23degeneracy} and \\ref{sensitivity}. \nNonetheless, the property makes the structure of analysis for resolving the \neight-fold degeneracy transparent, and therefore it may worth to report. \n\n\nThe problem of decoupling came to our attention via the following path. \nIn most part of the previous paper~\\cite{T2KK}, we have discussed \nhow to solve the sign-$\\Delta m^2$ degeneracy without worrying \nabout the $\\theta_{23}$ octant degeneracy. \nConversely, the authors of ~\\cite{resolve23} analyzed the latter \ndegeneracy without resolving the former one. \nAre these correct procedure? \nThe answer is yes if the analysis procedure and the results for \nthe $\\theta_{23}$ degeneracy is independent of which solutions \nwe take for the sign-$\\Delta m^2$ degeneracy, and vice versa. \nWe call this property the ``decoupling between the degeneracies''.\\footnote{\nHere is a concrete example for which the decoupling does not work; \nIn the method of comparison between \n$\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ disappearance measurement for \nlifting $\\theta_{23}$ octant degeneracy \\cite{choubey} one in fact \ndetermines the combined sign of \n$\\cos 2\\theta_{23} \\times \\Delta m^2_{31}$ as noticed in \\cite{resolve23}, \nand hence no decoupling.\n}\nThough discussion on this point was partially given in \\cite{resolve23}, \nwe present here a complete discussion of the decoupling.\n\n\nUnder the approximation of lowest nontrivial order in matter effect, \nwe prove that the decoupling holds between the above two degeneracies, \nand furthermore that it can be generalized, \nthough approximately, to the relation between any two pair of \ndegeneracies among the three types of degeneracies.\nTo our knowledge, leading order in matter perturbation theory appears to \nbe the only known circumstance that the argument goes through. \nFortunately, the approximation is valid for the setting used in this paper \nwith baseline up to $\\sim$1000 km, in particular in the Kamioka-Korea \nsetting. \nIn the following treatment we make a further approximation that the \ndegenerate solutions are determined primarily by the measurement at \nthe intermediate detector. \nIt is a sensible approximation because the statistics is about 10 times higher \nat the intermediate detector, and its validity is explicitly verified \nin the analysis performed in \\cite{T2KK}. \n\n\\subsection{Approximate analytic treatment of the parameter degeneracy}\n\n\nTo make the discussion self-contained, we start from the derivation of \nthe degenerate solutions by using the matter perturbation theory \\cite{AKS}, \nin which the matter effect is kept to its lowest nontrivial order.\nNamely, the matter effect can be ignored in leading order in \nthe disappearance channel whose oscillation probability is order of unity. \nThen, the disappearance probability \n$P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu})$ can be given by \nthe vacuum oscillation approximation with leading order in $s^2_{13}$ \nand the solar $\\Delta m^2_{21}$ corrections as \n\\begin{eqnarray}\n1 - P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu})  &=& \n\\left[ \n\\sin^2 2\\theta_{23}\n+ 4 s^2_{13} s^2_{23} \n\\left(2 s^2_{23} - 1 \\right)\n\\right]\n\\sin^2 \\left(\\frac{\\Delta m^2_{31}L}{4E}\\right) \n\\nonumber \\\\\n& - & c_{12}^2\\sin^22\\theta_{23}\n\\left(\\frac{\\Delta m_{21}^2 L}{4E}\\right) \n\\sin \\left(\\frac{\\Delta m_{31}^2 L}{2E}\\right).\n\\label{Pvac_mumu2}\n\\end{eqnarray}\nThe probability for anti-neutrino channel is the same as that for neutrino \none in this approximation. \nOne can show \\cite{AKS} that the other solar $\\Delta m^2_{21}$ \ncorrections are suppressed further by either a small \n$\\sin^2{2\\theta_{13}} \\lsim 0.1$, \nor the Jarlskog factor \n$J \\equiv c_{12} s_{12} c_{13}^2 s_{13} c_{23} s_{23} \\sin{\\delta} \\lsim 0.04$. \nIn fact, the validity of the approximation is explicitly verified in \\cite{resolve23} \nwhere the matter effect terms are shown to be of the order of $10^{-3}$ \neven at $L=1000$ km. \nA disappearance measurement, therefore, determines $s^2_{23}$ \nto first order in $s^2_{13}$ as \n\\begin{eqnarray}\n(s^2_{23})^{(1)}= (s^2_{23})^{(0)} (1+ s^2_{13}),\n\\label{degene-sol}\n\\end{eqnarray}\nwhere \n$(s^2_{23})^{(0)}$ is the solution obtained by ignoring $s^2_{13}$. \nFrom the first term in Eq.~(\\ref{Pvac_mumu2}), \nthe two solutions of $(s^2_{23})^{(0)}$ is determined as \n$(s^2_{23})^{(0)}= \\frac{1}{2} \\left[ 1 \\pm \\sqrt{1- \\sin^2{2\\theta_{23}}} \\right]$, \nthe simplest form of the $\\theta_{23}$ octant degeneracy. \nFor example, $(s^2_{23})^{(0)} = 0.4$ or 0.6 (0.45 or 0.55) for \n$\\sin^2{2\\theta_{23}}=0.96\\,(0.99)$. \n\n\nFor the appearance channel, we use the \n$\\nu_{\\mu} (\\bar{\\nu}_\\mu) \\to \\nu_{e} (\\bar\\nu_e$) \noscillation probability with first-order matter effect \\cite{AKS}\n\\begin{eqnarray}\nP[\\nu_{\\mu}(\\bar{\\nu}_{\\mu}) &\\rightarrow&  \n\\nu_{\\rm e}(\\bar{\\nu}_e)]  =  \nc^2_{23} \\sin^2{2\\theta_{12}} \n\\left(\\frac{\\Delta m^2_{21} L}{4 E}\\right)^2 \n\\nonumber \\\\\n&+& \\sin^2{2\\theta_{13}} s^2_{23}\n\\left[\n\\sin^2 \\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right)\n-\\frac {1}{2}\ns^2_{12}\n\\left(\\frac{\\Delta m^2_{21} L}{2 E}\\right)\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \n\\right.\n\\nonumber \\\\\n&&\\hspace*{22mm} {}\\pm\n\\left.\n\\left(\\frac {4 Ea}{\\Delta m^2_{31}}\\right)\n\\sin^2 {\\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right)}\n\\mp \n\\frac{aL}{2}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \n\\right]\n\\nonumber \\\\\n&+& \n2J_{r} \\left(\\frac{\\Delta m^2_{21} L}{2 E} \\right)\n\\left[\n\\cos{\\delta}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \\mp \n2 \\sin{\\delta}\n\\sin^2 \\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right) \n\\right], \n\\label{Pmue}\n\\end{eqnarray}\nwhere the terms of order \n$s_{13} \\left( \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}} \\right)^2$ \nand \n$aL s_{13} \\left( \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}} \\right)$ \nare neglected. \nIn Eq.~(\\ref{Pmue}), $a\\equiv \\sqrt 2 G_F N_e$  \\cite{MSW}\nwhere $G_F$ is the Fermi constant, $N_e$ denotes the averaged \nelectron number density along the neutrino trajectory in \nthe earth, \n$J_r$ $(\\equiv c_{12} s_{12} c_{13}^2 s_{13} c_{23} s_{23} )$ \ndenotes the reduced Jarlskog factor, and the upper and the \nlower sign $\\pm$ refer to the neutrino and \nanti-neutrino channels, respectively. \nWe take constant matter density approximation in this paper.\nThe first term of Eq.~(\\ref{Pmue}) is due to the oscillation driven by the \nsolar $\\Delta m^2_{21}$, which is essentially negligible in the intermediate \ndetector but not at the far detector and is of key importance to resolve the \n$\\theta_{23}$ octant degeneracy. \n\n\nWe make an approximation of ignoring terms of order \n$(\\Delta m^2_{21}\/\\Delta m^2_{31}) J_r \\cos{2\\theta_{23}}$ in \nEq.~(\\ref{Pmue}). \nNote that keeping only the leading order in this quantity \nis reasonable because $J_r \\lsim 0.04$, \n$|\\Delta m^2_{21}\/\\Delta m^2_{31}| \\simeq 1\/30$, and \n$\\cos{2\\theta_{23}}= \\pm 0.2$ for $\\sin^2{2\\theta_{23}} = 0.96$. \nThen, \nthe two degenerate solutions obey an approximate relationship \n\\begin{eqnarray}\n\\left( \\sin^2 {2\\theta_{13}} s^2_{23}  \\right)^{\\text{1st}} =\n\\left( \\sin^2 {2\\theta_{13}} s^2_{23}  \\right)^{\\text{2nd}}, \n\\label{1st_order}\n\\end{eqnarray}\nor, \n$s_{13}^{\\text{1st}}s_{23}^{\\text{1st}} \n= s_{13}^{\\text{2nd}}s_{23}^{\\text{2nd}}$\nignoring higher order terms in $s_{13}$. \nWe can neglect the leading order correction in $s^2_{13}$ to \n$s^2_{23}$ in these relations because it gives $O(s^4_{13})$ terms. \n\n\n\nAnalytic treatment of the intrinsic and the sign-$\\Delta m^2$ degeneracies \nis given in \\cite{MNP2}. \nIn an environment where the vacuum oscillation approximation applies the \nsolutions corresponding to the intrinsic degeneracy are given by \\cite{intrinsic} \n\\begin{eqnarray}\n\\theta_{13}^{(2)} = \\theta_{13}^{(1)}, \n\\hspace{1cm}\n\\delta^{(2)} = \\pi -  \\delta^{(1)},\n\\label{intrinsicDS}\n\\end{eqnarray}\nwhere the superscripts (1) and (2) label the solutions due to the \nintrinsic degeneracy. \nUnder the same approximation the solutions corresponding to the \nsign-$\\Delta m^2$ degeneracy are given by \\cite {MNjhep01}\n\\begin{eqnarray}\n\\theta_{13}^{\\,\\text{norm}} = \\theta_{13}^{\\,\\text{inv}}, \n\\hspace{1cm}\n\\delta^{\\,\\text{norm}} = \\pi - \\delta^{\\,\\text{inv}}, \n\\hspace{1cm}\n(\\Delta m^2_{31})^{\\,\\text{norm}} = - (\\Delta m^2_{31})^{\\,\\text{inv}}, \n\\label{signdm2DS}\n\\end{eqnarray}\nwhere the superscripts ``norm'' and ``inv'' label the solutions with \nthe positive and the negative sign of $\\Delta m^2_{31}$.  \nThe degeneracy stems from the approximate symmetry under the \nexchange of these two solutions through which the degeneracy is \nuncovered \\cite {MNjhep01}.\nThe validity of these approximate relationships  in the actual experimental \nsetup in the T2K II measurement is explicitly verified in \\cite{T2KK}. \nIt should be noticed that even if sizable matter effect is present \nthe relation (\\ref{intrinsicDS}) holds \nat the energy corresponding to the vacuum oscillation maximum, \nor more precisely, the shrunk ellipse limit \\cite{KMN02}. \n\n\n\n\\subsection{Decoupling between degeneracies}\n\n\nResolution of the degeneracy can be done when a measurement distinguishes \nbetween the values of the oscillation probabilities with the two different \nsolutions corresponding to a degeneracy. \nTherefore, we define the probability difference \n\\begin{eqnarray}\n&&\\Delta P^{ab}(\\nu_{\\alpha} \\rightarrow \\nu_{\\beta}) \n\\nonumber \\\\\n&\\equiv&  \nP \\left( \\nu_{\\alpha} \\rightarrow \\nu_{\\beta}; \\theta_{23}^{(a)}, \n\\theta_{13}^{(a)}, \\delta^{(a)}, (\\Delta m^2_{31})^{(a)} \\right) - \nP \\left( \\nu_{\\alpha} \\rightarrow \\nu_{\\beta}; \\theta_{23}^{(b)}, \n\\theta_{13}^{(b)}, \\delta^{(b)}, (\\Delta m^2_{31})^{(b)} \\right)\n\\label{DeltaPdef}\n\\end{eqnarray}\nwhere the superscripts $a$ and $b$ label the degenerate solutions. \nSuppose that we are discussing the degeneracy A. \nThe decoupling between the degeneracies A and B  \nholds if $\\Delta P^{ab}$ defined in (\\ref{DeltaPdef}) for the degeneracy A \nis invariant under the replacement of the mixing parameters \ncorresponding to the degeneracy B, and vice versa. \n\n\nThe best example of the decoupling is given by the one between \nthe $\\theta_{23}$ octant and the sign-$\\Delta m^2$ degeneracies. \nBy noting that \n$J_r^{\\,\\text{1st}} - J_r^{\\,\\text{2nd}} \n= \\cos 2\\theta_{23}^{\\,\\text{1st}} J_r^{\\,\\text{1st}}$ \nin leading order in $\\cos{2\\theta_{23}}$, \nthe difference between probabilities with the first and the second octant \nsolutions can be given by \n\\begin{eqnarray}\n&&\\Delta P^{\\,\\text{1st 2nd}}(\\nu_{\\mu} \\rightarrow \\nu_{e}) = \n\\cos{2\\theta_{23}^{\\,\\text{1st}}} \n\\sin^2{2\\theta_{12}} \n\\left(\\frac{\\Delta m^2_{21} L}{4 E}\\right)^2 \n\\nonumber \\\\\n&+&\n2J_{r}^{\\,\\text{1st}} \\cos 2\\theta_{23}^{\\,\\text{1st}} \n\\left(\\frac{\\Delta m^2_{21} L}{2 E} \\right)\n\\left[\n\\cos{\\delta}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \\mp \n2 \\sin{\\delta}\n\\sin^2 \\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right) \n\\right]. \n\\label{DeltaP_23}\n\\end{eqnarray}\nThe remarkable feature of (\\ref{DeltaP_23}) is that the leading-order \nmatter effect terms drops out completely. \nTherefore, our approximated treatment remains valid until the \nsecond order matter effect starts to become sizable in the \nappearance oscillation probability. \nMore importantly, $\\Delta P^{\\,\\text{1st~2nd}}$, being composed only of \nthe vacuum oscillation terms, \nis obviously invariant under the replacement \n$normal \\leftrightarrow inverted$ solutions with different signs of \n$\\Delta m^2_{31}$ given in Eq.~(\\ref{signdm2DS}). \nTherefore, the resolution of the $\\theta_{23}$ octant degeneracy \ncan be carried out without worrying about the presence of \nthe sign-$\\Delta m^2_{31}$ degeneracy. \n\n\n\nNext, we examine the inverse problem; \nDoes the determination of mass hierarchy decouple with \nthe resolution of the $\\theta_{23}$ degeneracy?\nOne can show, by using Eq.~(\\ref{Pmue}), that \nthe similar probability difference between the solutions in \nEq.~(\\ref{signdm2DS}) with the normal and the \ninverted hierarchies is given by \n\\begin{eqnarray}\n&&\\Delta P^{\\,\\text{norm~inv}} (\\nu_{\\mu} \\rightarrow \\nu_{e}) \n\\nonumber \\\\\n&=&\n\\sin^2{2\\theta_{13}^{\\,\\text{norm}}} (s_{23}^{\\,\\text{norm}})^2\n\\left[ \n- s^2_{12}\n\\left(\\frac{\\Delta m^2_{21} L}{2 E}\\right)\n\\sin \\left(\\frac{(\\Delta m^2_{31})^{\\,\\text{norm}} L}{2 E}\\right) \n\\right.\n\\nonumber \\\\\n\\hspace*{0mm} {}&\\pm&\n\\left.\n2(aL) \n\\left\\{\n \\left(\\frac{4E}{(\\Delta m^2_{31})^{\\,\\text{norm}} L}\\right) \n\\sin^2 \\left(\\frac{(\\Delta m^2_{31})^{\\,\\text{norm}} L}{4 E}\\right) -\n\\frac{1}{2} \\sin \\left(\\frac{(\\Delta m^2_{31})^{\\,\\text{norm}} L}{2 E}\\right) \n\\right\\}\n\\right]\n\\label{DeltaP_sign}\n\\end{eqnarray}\nwhere the superscripts ``norm'' and  ``inv'' can be exchanged \nif one want to start from the inverted hierarchy. \nWe notice that most of the vacuum oscillation terms, including the solar term, \ndrop out because of the invariance under \n$\\delta \\rightarrow \\pi - \\delta$ and \n$\\Delta m^2_{31} \\rightarrow - \\Delta m^2_{31}$. \nNow, we observe that $\\Delta P^{\\,\\,\\text{norm~inv}}$\nis invariant under the transformation \n$\\theta_{23}^{\\,\\text{1st}} \\leftrightarrow \\theta_{23}^{\\,\\text{2nd}}$ and \n$\\theta_{13}^{\\,\\text{1st}} \\leftrightarrow \\theta_{13}^{\\,\\text{2nd}}$, \nbecause $\\Delta P^{\\,\\,\\text{norm~inv}}$ depends upon \n$\\theta_{13}$ and $\\theta_{23}$ only through the combination \n$\\sin^2{2\\theta_{13}} s_{23}^{2}$ within our approximation. \nTherefore, the sign-$\\Delta m^2_{31}$ and the $\\theta_{23}$ \ndegeneracies decouple with each other. \n\n\nIn our previous paper, we have shown that the sign-$\\Delta m^2_{31}$ \ndegeneracy can be lifted by the Kamioka-Korea two detector setting. \nThe above argument for decoupling guarantees that our treatment \nis valid irrespective of the solutions assumed for $\\theta_{23}$ \ndegeneracy.\\footnote{\nWe remark that in most part of \\cite{T2KK}, we have assumed that \n$\\theta_{23}=\\pi\/4$ so that this problem itself does not exist. \nThe above discussion implies that even in the case of \nnon-maximal value of $\\theta_{23}$ the similar analysis \nas in \\cite{T2KK} must go through without knowing in which octant \n$\\theta_{23}$ lives. \nThe resultant sensitivity for resolving the mass hierarchy will change \nas $\\theta_{23}$ goes away from $\\pi\/4$ but only slightly \nas will be shown in Sec.~\\ref{sensitivity}. \n}\n\n\n\\subsection{Including the intrinsic degeneracy}\n\\label{intrinsic}\n\n\nNow we turn to the intrinsic degeneracy for which the situation \nis somewhat different. First of all, in our setting, \nthe intrinsic degeneracy is already \nresolved by spectrum informations at the intermediate detector \nif $\\theta_{13}$ is relatively large, $\\sin^2 2\\theta_{13} \\gsim 0.02$ \nas illustrated in Fig.~\\ref{intrinsicKamKorea} before the information \nfrom the far detector is utilized. \nIt means that there is no intrinsic degeneracy from the beginning \nin the analysis with spectrum informations.  \nBecause of this feature, the intrinsic degeneracy decouple from \nthe beginning from the task of resolving the eight-fold degeneracy in \nour setting for the relatively large values of $\\theta_{13}$.\nBecause of a further enhanced sensitivity, the intrinsic degeneracy \nmay be resolved to a smaller values of $\\theta_{13}$ in the Kamioka-Korea setting. \n\n\nIn fact, powerfulness of the spectral information for resolving \nthe intrinsic degeneracy can be understood easily by noting that \n$\\Delta P^{ab}$ defined in (\\ref{DeltaPdef}) is given by using the \nintrinsic degeneracy solution (\\ref{intrinsicDS}) as \n\\begin{eqnarray}\n&&\n\\Delta P^{12}(\\nu_{\\mu} \\rightarrow \\nu_{e}) =\n4 J_{r} \\left(\\frac{\\Delta m^2_{21} L}{2 E} \\right) \n\\cos{\\delta^{(1)}}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right),\n\\label{DeltaP_intrinsic}\n\\end{eqnarray}\nwith notable feature that the matter effect cancels out. \nNotice that even if the matter effect cannot be negligible the solution \n(\\ref{intrinsicDS}) holds for measurement at energies around the \nvacuum oscillation maximum. \nThe right-hand side of Eq.~(\\ref{DeltaP_intrinsic}) is proportional to $E^{-2}$ \nat energies near the vacuum oscillation maximum, \n$\\frac{\\Delta m^2_{21} L}{2 E} = \\pi$, and the steep energy \ndependence can be used to lift the degeneracy. \nHence, the spectrum analysis is a powerful tool for resolving the \nintrinsic degeneracy. \n\n\n\\subsection{The case that the intrinsic degeneracy is not solved} \n\n\nEven if $\\theta_{13}$ is too small, or if the energy resolution is too \npoor for the spectrum analysis to resolve the intrinsic degeneracy, \nwe can show that  the intrinsic degeneracy approximately \ndecouples from the other degeneracies. \n$\\Delta P^{12}$ in Eq.~(\\ref{DeltaP_intrinsic}) is not exactly but approximately \ninvariant under the transformation first-octant $\\leftrightarrow$ \nsecond-octant solutions. \nThe difference between $\\Delta P^{12} (\\text{1st})$ and \n$\\Delta P^{12} (\\text{2nd})$\nis of order $\\cos 2\\theta_{23} J_r \\Delta m^2_{21} \/ \\Delta m^2_{31} \n\\simeq 3 \\times 10^{-4}$ for $s^2_{23}=0.4$ and $\\sin^2 2\\theta_{13}= 0.1$\napart from the further suppression by \n$\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right)$ at around the \noscillation maximum. \nBeing the vacuum oscillation term $\\Delta P^{12}$ is obviously invariant \nunder the replacement $normal \\leftrightarrow inverted$ solutions \nwith different signs of $\\Delta m^2_{31}$. \nTherefore, resolution of the intrinsic degeneracy can be done, \nto a good approximation, independent of the presence of the \nsign-$\\Delta m^2_{31}$ and the $\\theta_{23}$ octant degeneracies. \n\n\nThe remaining problem we need to address is the inverse problem, \nwhether the resolution of the sign-$\\Delta m^2_{31}$ and the $\\theta_{23}$ \noctant degeneracies can be carried out without knowing solutions \nof the intrinsic degeneracy. \nThe sign-$\\Delta m^2_{31}$ degeneracy decouples from the intrinsic one \nbecause $\\Delta P^{\\,\\text{norm~inv}}$ in (\\ref{DeltaP_sign}) is invariant under \nthe exchange of two intrinsic degeneracy solutions. \nThe $\\theta_{23}$ octant degeneracy also approximately decouples \nfrom the intrinsic one. \n$\\Delta P^{\\,\\text{1st~2nd}}$ in (\\ref{DeltaP_23}) changes under the \ninterchange of two intrinsic degeneracy solutions only by the \nsame amount as the difference between $\\Delta P^{12}$ of \nthe first and the octant $\\theta_{23}$ solutions,  \n$\\cos 2\\theta_{23} J_r \\Delta m^2_{21} \/ \\Delta m^2_{31} \n\\simeq 3 \\times 10^{-4}$ (for $s^2_{23}=0.4$ and $\\sin^2 2\\theta_{13}= 0.1$). \n\n\nHere is a clarifying comment on what the decoupling really means; \nBecause of the cross-shaped structure of the degenerate solutions \nof the intrinsic and the sign-$\\Delta m^2_{31}$ degeneracies \n(as was shown in Sec.~\\ref{how}) \nthe decoupling of the former from the latter does {\\em not} imply that \nthe correct value of $\\delta$ can be extracted from the measurement \nwithout knowing the correct sign of $\\Delta m^2_{31}$.\\footnote{\nNotice, however, that it does {\\em not} obscure the CP violation, \nbecause the ambiguity is only two-fold; $\\delta \\leftrightarrow \\pi-\\delta$. \n}\nIt means that the elimination of one of the ``intrinsic'' degenerate pair \nsolutions related by $\\delta \\leftrightarrow \\pi-\\delta$ \nfor a given sign of $\\Delta m^2_{31}$ can be done without knowing \nthe mass hierarchy, the true sign of $\\Delta m^2_{31}$. \nTherefore, the situation that the intrinsic degeneracy is always \nresolved by the spectrum analysis in region of not too small \n$\\theta_{13}$, as is the case in our setting, is particularly transparent \none from this viewpoint. \n\n\nTo sum up, we have shown that to leading order in the matter effect  \nthe intrinsic, the sign-$\\Delta m^2_{31}$, and the $\\theta_{23}$ \noctant degeneracies decouples with each other. \nThey do so exactly except for between the intrinsic and the $\\theta_{23}$ \noctant degeneracies for which the decoupling is approximate but \nsufficiently good to allow one-by-one resolution of all the three types \nof degeneracies. \nThe decoupling implies that in analysis for lifting the eight-fold \ndegeneracy the structure of the $\\chi^2$ minimum is very simple in \nmulti-dimensional parameter space, and it may be of use in \ndiscussions of how to solve the degeneracy  \nin much wider context than that discussed in this paper.\n\n\n\\section{How identical two detector setting solves $\\theta_{23}$ octant degeneracy?}\n\\label{23degeneracy}\n\n\nNow, we turn to the problem of how the identical two detector \nsetting can resolve the $\\theta_{23}$ degeneracy, \nthe unique missing link in a program of resolving eight-fold \nparameter degeneracy in the Kamioka-Korea two detector setting. \nThe solar $\\Delta m^2$ oscillation term, \nthe first term in Eq.~(\\ref{Pmue}) with the coefficient of $c^2_{23}$, \nmay be of key importance to do the job. \nWhile it was argued on very general ground \\cite{resolve23} that\nthe $\\theta_{23}$ degeneracy is hard to resolve only by accelerator \nexperiments with baseline of $\\lsim 1000$ km or so,  \nthe argument can be circumvented if the solar term can be isolated. \nWe emphasize that the accuracy of the determination of $\\theta_{23}$ is \nseverely limited by the octant degeneracy, as discussed in detail \nin \\cite{MSS04}.\n\n\nTherefore, the question we must address first is the relative importance \nof the solar term to the remaining terms in $\\Delta P^{\\,\\text{1st~2nd}}$ \nin Eq.~(\\ref{DeltaP_23}). \nWe note that the ratio of the solar term to the $\\delta$-dependent \nsolar-atmospheric interference term in $\\Delta P^{\\,\\text{1st~2nd}}$ is \ngiven by \n$\\sin^2{2\\theta_{12}} (\\Delta m^2_{21} L \/ 4 E) \/ 4 J_{r}$, \nassuming the square parenthesis in (\\ref{DeltaP_23}) is of order unity. \nThe ratio is roughly given by \n$\\simeq 3 (1 \/ 30) (\\pi \/ 2) 0.86 (1\/ 4 J_{r} ) \\simeq 0.9~(0.16 \/ s_{13}) $ \nwith beam energy having the first oscillation maximum in Kamioka. \nTherefore, the solar term is indeed comparable or lager for\nsmaller $\\theta_{13}$ in size with the interference terms \nin $\\Delta P^{\\,\\text{1st~2nd}}$ at the far detector. \nObviously, the solar term is independent of $\\theta_{13}$,\nwhich suggests that the sensitivity to resolve the $\\theta_{23}$\ndegeneracy is almost independent of $\\theta_{13}$, \nas will be demonstrated in Sec.~\\ref{sensitivity}. \nWe note that while the solar term is the key to resolve \nthe $\\theta_{23}$ degeneracy, the interference terms \nalso contributes to lift the degeneracy. In particular, \nas shown in (\\ref{DeltaP_23}), \nthe $\\sin\\delta$ term has opposite sign when the polarity of the \nbeam is switched from the neutrino to the anti-neutrino runs. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{Patm_Psol_normal.eps}\n\\end{center}\n\\caption{\nThe energy dependence of the solar term (red solid line) is \ncontrasted with the ones of atmospheric plus interference \nterms in the $\\nu_{e}$ \nappearance oscillation probabilities with various values of CP phase \n$\\delta$; \n$\\delta=0$ (dotted line), \n$\\delta=\\pi\/2$ (dashed line), \n$\\delta=\\pi$ (dash-dotted line), and \n$\\delta=3\\pi\/2$ (double-dash-dotted line). \nFor this plot, we used analytic expression in Eq.~(\\ref{Pmue}); \n$P_{\\text{solar}}$ is defined to be the first term in \nEq.~(\\ref{Pmue}) whereas $P_{\\text{atm}}$ is defined to be \nthe rest in Eq.~(\\ref{Pmue}).  \n$\\bar{P}_{\\text{solar}}$ and $\\bar{P}_{\\text{atm}}$ refer to the \ncorresponding terms for anti-neutrinos. \n}\n\\label{solar-vs-atm}\n\\end{figure}\n\n\nThe next question we must address is how the solar term can be \nseparated from the other terms to have enhanced sensitivity to \nthe $\\theta_{23}$ degeneracy. \nTo understand the behavior of the solar term and its difference from \nthat of the atmospheric terms in the oscillation probability, \nwe plot in Fig.~\\ref{solar-vs-atm} a comparison between them \nin Kamioka (left panels) and in Korea (right panels) for various \nvalues of $\\delta$.\nAs one can observe in the right panels, the energy dependence \nof the solar oscillation term,  a monotonically decreasing \n(approximately $1\/E^2$) behavior with increasing energy, \nis quite different  from the oscillating behavior of the atmospheric ones. \nIt is also notable that the ratio of the solar term to the atmospheric-solar \ninterference term is quite different between the intermediate and \nthe far detectors.\nDue to the differing relative importance of the solar term in the two \ndetectors and the clear difference in the energy dependences \nbetween the solar and the atmospheric terms, \nthe spectrum analysis, the powerful method for resolving the \nintrinsic degeneracy, must be able to isolate the solar term \nfrom the remaining ones. \nThis will be demonstrated in the quantitative analysis in the next section.\n\n\nWe note that several alternative methods are proposed to resolve \nthe $\\theta_{23}$ degeneracy. \nThey include: \nthe atmospheric neutrino method \\cite{concha-smi_23,atm23,choubey2}, and \nthe reactor accelerator combined method \\cite{MSYIS,resolve23}, \nthe atmospheric accelerator combined method \\cite{atm-lbl}. \nThe atmospheric neutrino method discussed in \n\\cite{concha-smi_23,atm23} is closest to ours in physics \nprinciple of utilizing the solar mass scale oscillation effect. \nPossible advantage of the present method may be in a clean detection \nof the solar term by the intermediate-versus-far two detector comparison.\n\n\n\\section{Sensitivity for resolving  $\\theta_{23}$ octant degeneracy}\n\\label{sensitivity}\n\nIn this section, we describe details of our analysis for resolving \nthe $\\theta_{23}$ octant degeneracy. \nThey include treatment of experimental errors, treatment of \nbackground, and the statistical procedure which is used to \ninvestigate the sensitivity of the experiment. \nThen, the results of our analysis are presented. \n\n\n\\subsection{Assumptions and the definition of $\\chi^2$}\n\n\nIn order to understand the sensitivity of the experiment\nwith the two detector system at \n295~km (Kamioka) and 1050~km (Korea), we carry out \na detailed $\\chi^2$ analysis. \nTo address the $\\theta_{23}$ octant degeneracy, \nit is of course necessary to include $\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ \ndisappearance channels in addition to the appearance ones in our treatment. \nIn short, the definition of the statistical procedure is similar to the \none used in Ref.~\\cite{T2KK} with necessary extension for including \nmuon events. \nThe assumption on the experimental setting is also identical\nto that of the best performance setting identified in Ref.~\\cite{T2KK}.\nNamely, 0.27~Mton fiducial masses for the intermediate \nsite (Kamioka, 295~km) and the far site (Korea, 1050~km).\nThe neutrino beam is assumed to be 2.5 degree off-axis one produced\nby the upgraded J-PARC 4~MW proton beam. \nIt is assumed that the experiment will continue for 8 years\nwith 4 years of neutrino and 4 years of anti-neutrino runs. \n\nWe use various numbers and distributions available from\nreferences related to T2K  \\cite{JPARC-detail}, in which many of \nthe numbers are updated after the original proposal \\cite{T2K}.\nHere, we summarize the main assumptions and the methods \nused in the $\\chi^2$ analysis.\nWe use the reconstructed neutrino energy for single-Cherenkov-ring \nelectron and muon events.\nThe resolution in the reconstructed neutrino energy is 80~MeV \nfor quasi-elastic events. \nWe assume that \n$|\\Delta m_{31}^2|$ should be known precisely by the time when \nthe experiment we consider in this report will be carried out. \nWe take \n$\\Delta m_{31}^2 = \\pm 2.5 \\times 10^{-3} \\text{eV}^2$. \nHence, we assume that the \nenergy spectrum of the beam is the one expected by the\n2.5~degree off-axis-beam in T2K. The shape of the\nenergy spectrum for the anti-neutrino beam is assumed to be\nidentical to that of the neutrino beam. The event rate\nfor the anti-neutrino beam \nin the absence of neutrino oscillations\nis smaller by a factor of 3.4 due mostly to the lower neutrino \ninteraction cross sections and partly to the slightly lower \nflux. The signal to noise ratio is\nworse for the anti-neutrino beam than that for the neutrino beam\nby a factor of about 2. \n\n28 background electron events are expected \nfor the reconstructed neutrino energies between 350 and 850~MeV for\n$(0.75 \\times 0.0225 \\times 5)\\, \\text{MW} \\cdot \\text{Mton} \\cdot \\text{yr}$ \nmeasurement \nwith the neutrino beam. The energy dependence \nof the background rate and the rate itself are taken \nfrom \\cite{JPARC-detail}.\nThe background rate is expected to be higher in the lower \nneutrino energies. The expected number of electron events\nis assumed to be 122 for $\\sin^2 2\\theta_{13} =$0.1 with the \nsame detector exposure\nand beam, assuming the normal mass hierarchy and $\\delta = 0$.\n\n\nWe assume that the experiment is equipped with a near detector \nwhich measures the rate and the energy dependence of the background\nfor electron events,\nun-oscillated muon spectrum, \nand the signal detection efficiency.\nThese measurements are assumed to be carried out\nwithin the uncertainty of 5\\%. \nWe already demonstrated that the dependence on the \nassumed value of the experimental systematic errors is \nrather weak \\cite{T2KK}.\nWe stress that in the present setting \nthe detectors located in Kamioka and in Korea are not only \nidentical but also receive neutrino beams with essentially\nthe same energy distribution (due to the same off-axis angle of 2.5 degree) \nin the absence of oscillations. \nHowever, it was realized recently that, due to a non-circular shape \nof the decay pipe of the J-PARC neutrino beam line, \nthe flux energy spectra viewed at detectors in Kamioka and in \nKorea are expected to be slightly different even at the same off-axis angle, \nespecially in the high-energy tail of the spectrum \n\\cite{Rubbia-Meregaglia-Seoul2006}. \nThe possible difference between fluxes in the intermediate and the \nfar detectors is newly taken into account as a systematic error in \nthe present analysis.\n\n\nWe compute neutrino oscillation probabilities \nby numerically integrating neutrino evolution equation \nunder the constant density approximation.  \nThe average density is assumed to be 2.3 and 2.8 g\/cm$^3$ for the \nmatter along the beam line between the production target and \nKamioka and between the target and Korea, respectively \\cite{T2KK}.\nWe assume that the number of electron with respect to that of nucleons \nto be 0.5 to convert the matter density to the electron number density. \nIn our $\\chi^2$ analysis, \nwe fix the absolute value of $|\\Delta m_{31}^2|$ to be $2.5\\times 10^{-3}$ eV$^2$, \nand fix solar parameters as $\\Delta m_{21}^2 = 8\\times 10^{-5}$ eV$^2$ and \n$\\sin^2{\\theta_{12}}$=0.31.\n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{strategy.1.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{Examples of electron and muon events to be observed in \nKamioka and Korea for 4 years of neutrino plus \n4 years of anti-neutrino running \nare presented as a function of reconstructed neutrino energy. \nThe fiducial masses are taken to be 0.27~Mton for both the detectors in \nKamioka and Korea.\nThe dashed histograms for electron events show the background events. \nThe open circles show the expected\nenergy spectrum of signal events with \n$\\sin^2 \\theta_{23} =$0.40 and $\\sin^2 2 \\theta_{13} =$0.01.\nThe solid circles show the expected\nenergy spectrum of signal events with \n$\\sin^2 \\theta_{23} =$0.60 and $\\sin^2 2 \\theta_{13} =$0.0067.\nIn both cases, $\\delta = 3\\pi\/4$ and normal mass\nhierarchy are assumed in simulating the events. \n}\n\\label{fig:energy-spectrum-examples}\n\\end{figure}\n\n\nFig.~\\ref{fig:energy-spectrum-examples} shows an\nexample of the energy spectrum of electron and muon events\nto be observed in Kamioka and Korea for 4 years of\nneutrino beam plus 4 years of anti-neutrino beam.\nThe two sets of parameters give very similar spectrum \nfor both the electron and muon events at the Kamioka detector \nand the muon events at the Korean detector. \nHowever, due to the long baseline distance, the\nsolar term plays some role in the \n$^( \\overline{\\nu}^)_\\mu \\rightarrow\\, ^(\\overline{\\nu}^)_e$ \noscillation \nprobability at the Korean detector. Therefore, \nthe two sets of parameters give slightly different\noscillation probabilities in Korea. \nSince the solar term is proportional to $c^2_{23}$ \nwe use this feature to obtain information on\n$\\sin^2 \\theta_{23}$.\n\n\nThe statistical significance of the measurement considered in this\npaper was estimated by using the following \ndefinition of $\\chi^2$:\n\\begin{equation}\n\\chi^2 =  \\sum_{k=1}^{4} \\left( \n\\sum_{i=1}^{5}\n\\frac{\\left(N(e)_{i}^{\\rm obs} - N(e)_{i}^{\\rm exp}\\right)^2}\n{ \\sigma^2_{i} } +\n\\sum_{i=1}^{20}\n\\frac{\\left(N(\\mu)_{i}^{\\rm obs} - N(\\mu)_{i}^{\\rm exp}\\right)^2}\n{ \\sigma^2_{i} }\n\\right)\n+ \\sum_{j=1}^{7} \\left(\\frac{\\epsilon_j}\n{\\tilde{\\sigma}_{j}}\\right)^2, \n\\label{equation:chi2def}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n N(e)_{i}^{\\rm exp} = N_{i}^{\\rm BG} \\cdot \n   (1+\\sum_{j=1,2,7} f(e)_{j}^{i}\\cdot\\epsilon_{j}) \n    + N_i^{\\rm signal} \\cdot \n   (1+\\sum_{j=3,7} f(e)_{j}^{i}\\cdot\\epsilon_{j}) ~,\n\\label{equation:e-number}\n\\\\\n N(\\mu)_{i}^{\\rm exp} = N_{i}^{\\rm non-QE} \\cdot \n   (1+\\sum_{j=4,6,7} f(\\mu)_{j}^{i}\\cdot\\epsilon_{j}) \n    + N_i^{\\rm QE} \\cdot \n   (1+\\sum_{j=4,5,7}   f(\\mu)_{j}^{i}\\cdot\\epsilon_{j}) ~.\n\\label{equation:mu-number}\n\\end{eqnarray} \n\n\\noindent\nThe first and second terms in Eq.~(\\ref{equation:chi2def}) are\n for the number of observed \nsingle-ring electron and muon events, respectively.\n$N(e~{\\rm or}~\\mu)^{\\rm obs}_i$ is the number of events to be \nobserved for the given oscillation parameter set,\n and $N(e~{\\rm or}~\\mu )^{\\rm exp}_i$ is the expected number of\nevents for the assumed oscillation parameters \nin the $\\chi^2$ analysis. \n$k=1,2,3$ and $4$ correspond to the four combinations \nof the detectors in Kamioka and in Korea with the \nneutrino and anti-neutrino beams,\nrespectively.\nThe index $i$ represents the reconstructed neutrino energy bin \nfor both electrons and muons. \nFor electron events, both $N(e)^{\\rm obs}_i$ and  $N(e)^{\\rm exp}_i$ \ninclude background events.\nThe energy ranges of the five energy bins for electron events \nare respectively\n400-500~MeV, \n500-600~MeV, 600-700~MeV, 700-800~MeV and \n800-1200~MeV. \nThe energy range for the muon events covers \nfrom 200 to 1200~MeV. Each energy bin has 50~MeV width.\n$\\sigma_i$ denotes the\nstatistical uncertainties in the expected data. \nThe third term  \nin the $\\chi^2$ definition collects the\ncontributions from variables which parameterize the systematic\nuncertainties in the expected number of signal and background events. \n\n\n$N^{\\rm BG}_i$ is the number of background events\nfor the $i^{\\rm th}$ bin for electrons.\n$N^{\\rm signal}_i$ is the number of electron appearance events\nthat are observed, and depends on \nneutrino oscillation parameters.\nThe uncertainties in $N^{\\rm BG}_i$ and $N^{\\rm signal}_i$\nare represented by 4 parameters $\\epsilon_j$ ($j=1$ to 3 and 7).\nSimilarly, $N^{\\rm non-QE}_i$ are the number of non-quasi-elastic events\nfor the $i^{\\rm th}$ bin for muons. \n$N^{\\rm QE}_i$ are the number of quasi-elastic muon events.\nWe treat the non-quasi-elastic and quasi-elastic muon events separately,\nsince the neutrino energy cannot be properly reconstructed for\n non-quasi-elastic events.\nBoth  $N^{\\rm non-QE}_i$ and $N^{\\rm QE}_i$ depend on \nneutrino oscillation parameters.\nThe uncertainties in  $N^{\\rm non-QE}_i$ and $N^{\\rm QE}_i$  \nare represented by 4 parameters $\\epsilon_j$ ($j=4$ to 7). \n\n\nDuring the fit, the values of $N(e~{\\rm or}~\\mu)^{\\rm exp}_i$ are\nrecalculated \nfor each choice of the oscillation parameters which are varied freely \nto minimize $\\chi^2$, and so are the systematic error parameters \n$\\epsilon_j$. \nThe parameter $f(e~{\\rm or}~\\mu)^i_j$ represents \nthe fractional change in the predicted\nevent rate in the $i^{\\rm th}$ bin due to a variation of the parameter\n$\\epsilon_j$. \nThe overall background normalization\nfor electron events \nis assumed to be uncertain by $\\pm$5\\% ($\\tilde{\\sigma}_{1}$=0.05). \n      It is also assumed that the background events for electron\n      events have an energy dependent uncertainty with the functional\n      form of $f(e)_2^i=((E_\\nu(rec)-800~\\text{MeV}) \/ 400~\\text{MeV})$. \n      5\\% is assumed to be the\n      uncertainty in $\\epsilon_2$ ($\\tilde{\\sigma}_{2}$=0.05).\n      The functional form of\n      $f(\\mu)_4^i = (E_\\nu(rec)-800~\\text{MeV}) \/ 800~\\text{MeV}$\n      is used to define the uncertainty in the spectrum shape for\n      muon events ($\\tilde{\\sigma}_{4}$=0.05).\nThe uncertainties in the signal detection efficiency \nare assumed to be 5\\% for both electron and muon events\n($\\tilde{\\sigma}_{3}=\\tilde{\\sigma}_{5}$=0.05).\nThe uncertainty in the separation of quasi-elastic and \nnon-quasi-elastic interactions in the muon events is\nassumed to be 20\\% ($\\tilde{\\sigma}_{6}$=0.20).\nThese systematic errors are assumed to be not correlated between\nthe electron and muon events.\nIn addition, for the number of events in Korea, the possible \nflux difference between Kamioka and Korea is taken into account\nin $f(e~{\\rm or}~\\mu)_7^i$. The predicted flux \ndifference \\cite{Rubbia-Meregaglia-Seoul2006} \nis simply assumed to be the 1~$\\sigma$ uncertainty in the flux \ndifference ($\\tilde{\\sigma}_{7}$).\n\n\n\\subsection{Sensitivity with two-detector complex}\n\n\nNow we present the results of the sensitivity analysis for \nthe $\\theta_{23}$ octant degeneracy. \nThe results for the mass hierarchy as well as CP violation sensitivities \nwill be discussed in the next section.\nFig.~\\ref{sensitivity-theta23-octant} shows\nthe sensitivity to the $\\theta_{23}$ octant determination as a function of \n$\\sin^2 2 \\theta_{13}$ and $\\sin^2 \\theta_{23}$. \nThe areas shaded with light (dark) gray of this figure indicate \nthe regions of parameters where the octant of $\\theta_{23}$ \ncan be determined at 2 (3) standard deviation confidence level, \nwhich is determined by the condition \n$\\chi^2_{\\text{min}} \\text{(wrong~octant)}\n-\\chi^2_{\\text{min}}\\text{(true~octant)}>$ 4 (9). \nThe upper (lower) panels correspond to the case where\nthe true hierarchy is normal (inverted). \nNote, however, that the fit was performed without assuming \nthe mass hierarchy. \nSince the sensitivity mildly depends on the CP phase $\\delta$, \nwe define the sensitivity to resolving the octant degeneracy \nin two ways: \nthe left (right) panels correspond to the case where \nthe sensitivity is defined such that the octant is determined \nfor any value of delta (half of the $\\delta$ space). \nFrom this figure, we conclude that the experiment \nwe consider here is able to solve the octant ambiguity,\nif $\\sin^2 \\theta_{23} < 0.38\\,(0.42) $ or $>0.62\\,(0.58)$ at 3\n(2) standard deviation confidence level. This conclusion depends\nweakly on the value of $\\sin^2 2 \\theta_{13}$, as well \nas the value of the CP phase $\\delta$ and the mass hierarchy.\n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{region2.1.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{ 2 (light gray area) and 3 (dark gray area) \nstandard deviation sensitivities to the $\\theta_{23}$\noctant degeneracy for 0.27~Mton detectors both in Kamioka\nand Korea.\n4 years running with neutrino beam and another 4 years with \nanti-neutrino beam are assumed.\nIn (a), the sensitivity is defined so that the experiment \nis able to identify the octant of $\\theta_{23}$ for any \nvalues of the CP phase $\\delta$. In (b), it is defined so that the \nexperiment is able to identify the octant of $\\theta_{23}$ \nfor half of the CP $\\delta$ phase space.\n}\n\\label{sensitivity-theta23-octant}\n\\end{figure}\n\n\nThe sensitivity of lifting the octant degeneracy by this setting \nis quite high even for rather small values of $\\theta_{13}$ to \n$\\sin^2 2 \\theta_{13} \\sim 10^{-3}$ where the mass hierarchy is \nnot determined, a possible consequence of the decoupling. \nSee Figs.~\\ref{sensitivity-mass-hierarchy} in the next section. \nThe sensitivity depends very weakly on $\\theta_{13}$ \nin relatively small values of \n$\\sin^2 2\\theta_{13}$ where the dominant atmospheric terms \nare small.\nThe feature of almost independence of the sensitivity to $\\theta_{13}$ \nshould be contrasted with that of the accelerator-reactor combined \nmethod in which a strong dependence on $\\theta_{13}$ \nis expected \\cite{resolve23}. \nVery roughly speaking the sensitivity by the present method is \nbetter than the latter method in a region \n$\\sin^2 2 \\theta_{13} \\lsim 0.05-0.06$ \naccording to the result given in Fig.~8 of \\cite{resolve23}. \nThe sensitivity of our method is also at least comparable to that could \nbe achieved by the high statistics observation of atmospheric neutrinos \n\\cite{concha-smi_23,atm23,choubey2}. \n\n\n\\section{Reexamination of sensitivities to neutrino mass hierarchy and CP violation} \n\\label{revisit}\n\n\nIn this section we reexamine the problem of sensitivities \nto the neutrino mass hierarchy and CP violation \nachievable by the Kamioka-Korea identical two detector complex.  \nWe want to verify that the sensitivities do not depend on \nwhich octant $\\theta_{23}$ lives, as indicated by our  \ndiscussion of the decoupling given in Sec.~\\ref{decoupling}. \nIt is also interesting to examine how the \nsensitivities depend upon $\\sin ^2 2\\theta_{23}$. \nFurthermore, the inclusion of the new systematic error which accounts \nfor difference in the spectral shapes of the neutrino beam between \nthe intermediate and the far detectors makes the reexamination \nworth to do. \n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.73\\textwidth]{region.1.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{ 2(thin lines) and 3(thick lines) \nstandard deviation sensitivities to the mass hierarchy \ndetermination for\nseveral values of $\\sin ^2 2 \\theta_{23}$ (red, yellow, \nblack, green and blue lines show the results for \n$\\sin^2 \\theta_{23} =$ 0.40, 0.45, 0.50, 0.55 and 0.60, \nrespectively). The sensitivity is defined in the plane\nof $\\sin ^2 2 \\theta_{13}$ versus CP phase $\\delta$.\nThe top and bottom panels show the cases for positive and \nnegative mass hierarchies, respectively. The experimental\nsetting is identical to that in \nFig.\\ref{sensitivity-theta23-octant}. \n}\n\\label{sensitivity-mass-hierarchy}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.73\\textwidth]{region.2.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{ Sensitivities to the CP violation, \n$\\sin \\delta \\ne0$. \nThe meaning of the lines \nand colors are identical to that in \nFig.~\\ref{sensitivity-mass-hierarchy}.\n}\n\\label{sensitivity-CP}\n\\end{figure}\n\n\nIn Figs.~\\ref{sensitivity-mass-hierarchy} and \\ref{sensitivity-CP} \nthe regions sensitive to the mass hierarchy and CP violation, \nrespectively, are presented. \nIn both figures, the thin-lines and the thick-lines \nindicate the sensitivity region at 2 and 3 standard deviations, \nrespectively.\nAs in the previous work~\\cite{T2KK}, 2 (3) standard deviation \nsensitivity regions are defined by the conditions, \n$\\chi^2_{\\text{min}} \\text{(wrong~hierarchy)}-\\chi^2_{\\text{min}}\\text{(true~hierarchy)}>$\n4 (9) and \n$\\chi^2_{\\text{min}} (\\delta = 0\\ \\text{or}\\ \\pi)\n-\\chi^2_{\\text{min}}(\\text{true value of}\\, \\delta)>$\n4 (9) for the mass hierarchy and CP violation, respectively. \n\n\nThe sensitivities to the mass hierarchy and CP violation \nat $\\sin ^2 \\theta_{23} = 0.5$ are almost identical to those obtained in \n\\cite{T2KK}. \nIt is evident that the sensitivities do not depend strongly on \n$\\sin ^2 \\theta_{23}$ as far as the value is between 0.40 and 0.60.\nIn fact, the mass hierarchy can be determined even if \nthe $\\theta_{23}$ octant degeneracy is not resolved. \nBut, the sensitivity to mass hierarchy resolution gradually improves \nas $\\sin ^2 \\theta_{23}$ becomes larger, as seen in \nFig.~\\ref{sensitivity-mass-hierarchy}. \nIt is natural because $\\Delta P^{\\,\\text{norm~inv}}$ in (\\ref{DeltaP_sign}), \nor the appearance probability itself is proportional to $\\sin ^2 \\theta_{23}$. \nAn alternative way of presenting the same result is to use \n$s^2_{23} \\sin ^2 2\\theta_{13}$ for the ordinate. \nAn approximate scaling behavior is observed as expected \nby $\\Delta P^{\\,\\text{norm~inv}}$ in (\\ref{DeltaP_sign}).\n\n\n\n\\section{Summary and Discussion}\n\\label{summary}\n\n\nIn this paper, we have shown that a setting with two identical water \nCherenkov detectors of 0.27 Mton fiducial mass, one in Kamioka \nand the other in Korea, which receive almost the same neutrino beam \nfrom J-PARC has capability of resolving the $\\theta_{23}$ \noctant degeneracy {\\em in situ} by observing difference of the \nsolar oscillation term between both detectors. \nThe feature of the sensitivity region indicates that \nthe present method is quite complementary to the reactor-accelerator \ncombined method explored in \\cite{resolve23}. \nTogether with the potential for resolution of the intrinsic and the \nsign-$\\Delta m^2_{31}$ degeneracies previously reported in \\cite{T2KK} \n(with confirmation in Sec.~\\ref{revisit} by an improved treatment),  \nwe have demonstrated that the Kamioka-Korea two detector complex \ncan resolve all the eight-fold neutrino parameter degeneracy \nunder the assumption that $\\theta_{13}$ is within reach by the next \ngeneration accelerator experiments and $\\theta_{23}$ \nis not too close to $\\pi\/4$. \n\n\nAs an outcome of these studies, the strategy toward determination \nof the remaining unknowns in the lepton flavor mixing can be\ndiscussed. \nIt is nice to see that such program can be defined only with the single \nexperiment based on the conventional superbeam \ntechnology which does not require long-term R$\\&$D efforts,  \nand the well established detector technology.\nIt opens the possibility of accurate determination of the neutrino mixing \nparameters, $\\theta_{23}$, $\\theta_{13}$, $\\delta$, \nas well as the neutrino mass hierarchy, \nby lifting all the eight-fold degeneracy  \nwhich should merit our understanding of physics of lepton sector. \n\n\nOur treatment in this paper includes a new systematic error which \naccounts for possible difference in spectral shape of the neutrino beam \nreceived by the two detectors in Kamioka and in Korea.  \nWe have shown that, despite the existence of such new uncertainty \nwhich might hurt the principle of near-far cancellation of the \nsystematic errors, the capability of determining neutrino mass hierarchy \nand sensitivity to CP violation are kept intact. \n\n\nWe have also reported a progress in understanding the theoretical \naspect of the problem of how to solve the parameter degeneracy. \nBecause of the property phrased as ``decoupling between degeneracies'' \nwhich is shown to hold in a setting that allows perturbative treatment of \nmatter effect, \none can try to solve a particular degeneracy without worrying \nabout the presence of other degeneracies. \nThis feature may be contrasted to those \nof the very long baseline approaches, such as the neutrino factory, \nin which one would not expect the discussion in this paper to \nhold.\n\n\nAn alternative but closely related approach toward determination \nof the global structure of lepton flavor mixing in a single experiment \nis to utilize an on-axis wide band neutrino beam to explore the \nmultiple oscillation maxima, which may be called the ``BNL strategy'' \n\\cite{BNL,FNALversion}. \nThis strategy can be applied to the far detector in Korea, as examined \nby several authors \\cite{Hagiwara,Dufour-Seoul2006,Rubbia-Seoul2006}.\\footnote{\nVery roughly speaking ignoring the issue of backgrounds\nand assuming the same baseline length, \none would expect that wide band beam option is better in sensitivity \nto the neutrino mass hierarchy, \nwhile the same off-axis angle option studied in this paper is \nadvantageous to resolve the $\\theta_{23}$ octant degeneracy \nfor which low energy bins are essential.\n}\nIn this case, however, one needs to understand the energy \ndependence of the background and the signal efficiency \nas well as the neutrino interaction cross section precisely \nfor both the intermediate and the far detectors.\nIn particular, since the low energy bins are enriched with \nneutral current background contamination that comes from \nevents with higher neutrino energies \n\\cite{Dufour-Seoul2006} \nthe cancellation of the systematic errors between \nthe two detectors, which is the key ingredient in our analysis, \ndoes not hold. \nNonetheless, we emphasize that the potentially powerful method is \nworth to examine further with realistic estimate of the \ndetector performance.\n\n\nFinally, we remark that the J-PARC 2.5 degree off-axis beam\nwith the baseline length of 1,000 to 1,250~km should be \navailable in the Korean Peninsula. Therefore, it may be possible to\nfurther enhance the sensitivity to the $\\theta_{23}$ octant\nby taking a longer baseline length for the Korean detector.\nThe best baseline length and the detector location \nshould be decided so that the experiment has the best \nsensitivities to the oscillation parameters, especially\nto the CP phase $\\delta$, mass hierarchy and the octant of \n$\\theta_{23}$. \n\n\n\\begin{acknowledgments}\nWe would like to thank M.~Ishitsuka and K.~Okumura for \nthe assistance in the analysis code. \nH.N. thanks Stephen Parke and Olga Mena for useful discussion. \nThis work was supported in part by the Grant-in-Aid for Scientific Research, \nNos. 15204016 and 16340078, Japan Society for the Promotion of Science, \nFunda\\c{c}\\~ao de Amparo \\`a Pesquisa do Estado de Rio de Janeiro (FAPERJ) \nand by Conselho Nacional de Ci\\^encia e  Tecnologia (CNPq). \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe hierarchical model of structure formation predicts that clusters\nof galaxies form by subsequent merging of smaller structures. A\nconsistent amount of energy ($\\sim 10^{63} - 10^{64}$ ergs) is\nreleased in the Intra Cluster Medium (ICM) as result of such merger\nevents. Cosmological numerical simulations have shown that shocks and\nturbulence associated with these processes do not only heat the ICM\nbut also play an important role in non-thermal phenomena occurring in\nthe ICM (see e. g. the review by Dolag et al. 2008). Although these\nprocesses are not completely understood yet, this energy input is\nsupposed to accelerate and inject relativistic particles on cluster\nscale and amplify the magnetic field in the ICM.\\\\ Extended radio\nsources on cluster scale not associated with any optical counterpart\nbut arising from the ICM have been detected in an increasing number of\ngalaxy clusters. They are called radio halos and radio relics\ndepending on their morphology, location and radio properties. \\\\ Radio\nrelics are extended radio sources located at the outskirts of galaxy\nclusters, and are strongly polarized, with linear fractional\npolarization at 20 cm above 10 $\\%$, reaching values up to 50 $\\%$ in\nsome regions (see e.g. Govoni \\& Feretti 2004; Ferrari et\nal. 2008). Their origin is debated and not well known.  There is a\ngeneral consensus that it is related to phenomena occurring in the ICM\nduring merging events. So far, there are $\\sim$ 20 clusters of\ngalaxies where at least one radio relic is present. Their radio\nmorphology and location are quite varied, and could reflect different\nphysical origin or ICM conditions (Kempner et al. 2004; Giovannini \\&\nFeretti 2004). Due to low X-ray brightness at the cluster periphery, a\ncomparison of relic properties with the surrounding medium\n(i. e. temperature and brightness gradient induced by shock waves) is\nnot obvious. Feretti \\& Neumann (2006) did not find any evidence of a\ntemperature jump nearby the Coma cluster relic, and only recently a\ntemperature gradient has been found nearby the relic in Abell 548b by\nSolovyeva et al. (2008); here a precise orientation of the cluster\nmerger with respect to the line of sight is inferred, and the\nprojected displacement of the relic from the shock is thus\nexplained.\\\\ Of particular interest is to explore the connection between\nmerger shock waves and clusters with double relics,\ni.e. clusters hosting two relic radio sources located in the\nperipheral region and symmetric with respect to the cluster center. So\nfar a very small number of clusters with two double relics has been\nfound. One of them is Abell 3667 (R\\\"ottgering et al. 1997;\nJohnston-Hollitt et al. 2002). Here the cluster X-ray emission shows\nan elongated shape, interpreted as the merger axis of two\nsub-clusters, and relics are displaced symmetrically and perpendicular\nto the main axis. X-ray, optical and radio properties have been\nreproduced by a numerical simulation of a merger between clusters with\nmass ratio of 0.2 by Roettiger et al. (1999). We note however that not\nall of the predictions made by such simulations could be tested with\navailable data. Apart from Abell 3667, double relics have been\nobserved in Abell 3376 (Bagchi et al. 2006), and interpreted as\n``Outgoing merger shock waves''. Double relics have also been observed\nin RXCJ 1314.4-2515 (Feretti et al. 2005; Venturi et al. 2007), but no\ndetailed study on the relics formation has been performed on this\ncluster so far. Two more candidates for hosting double relics are\nAbell 2345 (Giovannini et al. 1999) and Abell 1240 (Kempner \\& Sarazin\n2001).\\\\\n\\begin{table*} [t!]\n\\caption{VLA observations}          \n\\label{tab:radioobs}      \n\\centering          \n\\begin{tabular}{|c c c c c c c c|}    \n\\hline\\hline       \nSource       & RA           &  DEC      &$\\nu$&Bandwidth&Config.& Date & Duration     \\\\\n             & (J2000)      & (J2000)   &   (MHZ)  & (MHz)   &       &      & (Hours)  \\\\\\hline                    \n\\hline\nAbell 2345   & 21 27 12.0   & -12 10 30.0 &   325    & 3.125   &   B    &16-AUG-2006 & 2.0 \\\\% 147779\\\\ \n             &              &             &   325    & 3.125   &   C    &08-DEC-2006 & 5.4 \\\\%263514\\\\\nAbell 1240   & 11 23 37.0   & 43 05 15.0  &   325    & 3.125   &   B    &05-AUG-2006 & 2.6\\\\% 214384\\\\\n             &              &             &   325    & 3.125   &   C    &08-DEC-2006 & 4.7 \\\\%203675\\\\\n\\hline\nAbell 2345-1 & 21 26 43.0  & -12 07 50.0 &    1425    & 50      &   C    &08-DEC-2006 & 1.9 \\\\%185755 \\\\%(24)\n             & 21 26 43.0  & -12 07 50.0 &    1425    & 50      &   D    &09-APR-2007 & 1.0 \\\\%98376 \\\\%(24)\nAbell 2345-2 & 21 27 36.0  &-12 11 25.0 &     1425    & 50      &   C    &08-DEC-2006 & 2.0 \\\\%196021\\\\%(24)\n             & 21 27 36.0  &-12 11 25.0 &     1425    & 50      &   D    &09-APR-2007 & 1.0 \\\\%108368\\\\%(24)\nAbell 1240-1 & 11 23 25.0  & 43 10 30.0 &     1425    & 50      &   C    &08-DEC-2006& 1.8 \\\\% 177879\\\\\n             & 11 23 25.0  & 43 10 30.0 &     1425    & 50      &   D    &12-APR-2007& 1.0 \\\\%80364 \\\\\nAbell 1240-2 & 11 23 50.0 & 43 00 20.0  &     1425    & 50      &   C    &08-DEC-2006& 1.9 \\\\%189945 \\\\\n             & 11 23 50.0 & 43 00 20.0  &     1425    & 50      &   D    &12-APR-2007& 1.0 \\\\%78099\\\\                  \n\\hline\nAbell 2345   & 21 26 57.2 & -12 12 49   &    1490     & 50      &  AnB   &02-NOV-1991& 0.1\\\\\n\\hline\n\\multicolumn{8}{l}{\\scriptsize Col. 1: Source name; Col. 2, Col. 3: Pointing position (RA, DEC);\nCol. 4: Observing frequency;}\\\\\n\\multicolumn{8}{l}{\\scriptsize Col 5: Observing bandwidth; Col. 6: VLA configuration; \nCol. 7: Dates of observation; Col. 8: Net time on source.}\\\\ \n\\end{tabular}\n\\end{table*}\nWe present here\nnew Very Large Array (VLA) observations of these two clusters at 20\nand 90 cm to confirm and study the double relic emission in the\nframework of relic formation models. Spectral index analysis of both\nradio relics in the same cluster have not been performed so far. In\nAbell 3667 the spectral index image has been obtained for only one of\nthe two relics, and no spectral index information are available for\nrelics in Abell 3376. Only integrated spectral index information are\navailable for the relics in RXCJ 1314.4-2515. Study of the spectral\nindex and of the polarization properties of relics offers a powerful\ntool to investigate the connection between double relics and outgoing\nshock waves originating in a merger event. In fact, theoretical models\nand numerical simulations make clear predictions on the relic spectral\nindex trend and magnetic field properties (see Ensslin et al.\n1998; Roettiger et al. 1999; Hoeft \\& Br{\\\"u}ggen 2007).\\\\\nThe paper is\norganized as follows: in Sec. 2 observations and data reduction are\ndescribed, in Sec. 3 and 4 we present the analysis of the cluster\nAbell 2345 and Abell 1240.  Results are discussed in Sec. 5 in\ncomparison with other observed relics and theoretical models, and\nconclusions are presented in Sec. 6. We assume a $\\Lambda$CDM\ncosmological model with $H_0=$71 km s$^{-1}$ Mpc$^{-1}$, $\\Omega_M$=0.27,\n$\\Omega_{\\Lambda}$=0.73.\n\n\\section{VLA radio observations}\n\\subsection{Total intensity data reduction}\n\\label{sec:radioobs}\nObservations have been performed at the Very Large Array (VLA) at 20\ncm in the C and D configuration and at 90 cm in the B and C\nconfiguration, in order to obtain the same spatial frequency coverage\nin the UV plane. Observations details are given in\nTab. \\ref{tab:radioobs}. \\\\ {\\bf Observations at 20 cm (1.4 GHz)} have\nbeen pointed separately on the two relics in both of the clusters\nbecause of the smaller full width at half power of the primary\nbeam. Observations of the cluster Abell 1240 have been calibrated\nusing the source 3C286 as primary flux density calibrator\\footnote{we\nrefer to the flux density scale by Baars \\& Martin (1990)}. The source\n1156+314 has been observed at intervals of about 30 min and used as\nphase calibrator. Observations of Abell 2345 have been calibrated\nusing the sources 3C48 as primary flux density calibrator. Phase\ncalibration has been performed by observing the source 2137-207 at\nintervals of $\\sim$ 30 min.\\\\ We performed standard calibration and\nimaging using the NRAO Astronomical Imaging Processing Systems\n(AIPS). Cycles of phase self-calibration were performed to refine\nantennas phase solutions, followed by a final amplitude and gain\nself-calibration cycle.\\\\ In addition we recovered from the VLA data\narchive a short observation performed with AnB array. The source 3C48\nis used as primary flux density calibrator and the source 2121+053 is\nused as phase calibrator. We reduced and calibrated these data as\nexplained above, details are given in\nTab. {\\ref{tab:radioobs}}.  \\\\{ \\bf Observations at 90 cm (325 MHz)}\nhave been performed in the spectral line mode, using 32 channels with\n3.127 MHz bandwidth. This observing method avoids part of the VLA\ninternal electronics interferences and allows us to remove accurately\nRadio Frequency Interferences (RFI). This also reduces bandwidth\nsmearing, that is quite strong at low frequencies. Primary flux\ndensity and phase calibrators were the same sources used in 1.4 GHz\nobservations. 3C48 and 3C286 were also used for bandpass\ncalibration. RFI are particularly strong at low radio frequency, so\nthat an accurate editing has been done channel by channel, resulting\nin a consistent flag of data. This in conjunction with bad data coming\nfrom EVLA antennas results in a loss of $\\sim$ 40 \\% of observing\ntime.  Calibration has been performed following the ``Suggestions for\nP band data reduction'' by Owen et al. (2004).\\\\ After the initial\nbandpass calibration channels from 1 to 4 and from 28 to 32 have been\nflagged because of the roll-off of the bandpass. In the imaging\nprocedure data have been averaged to 8 channels. Imaging has been\nperformed using the wide field imaging technique to correct for non\ncomplanarity effects over a wide field of view. 25 facets covering\nthe main lobe of the primary beam have been used in the cleaning and\nphase-self calibration processes. We also searched in\nthe NVSS data archive for sources stronger then 0.5 Jy over a radius\nas large as 10$^{\\circ}$. These sources have been included in the\ninitial cleaning and self calibration steps.\\\\ Each (u,v) data set at\nthe same frequency but observed with different configurations has been\ncalibrated, reduced and imaged separately and then combined to produce\nthe final images. Images resulting from the separate pointed\nobservations at 1.4 GHz have been then linearly combined with the AIPS\ntask LTESS. We combined the data set and produced images at higher and\nlower resolution (herein after HR images and LR images) giving uniform\nand natural weight to the data. For the purposes of the spectral\nanalysis, the final images at 325 MHz and 1.4 GHz, have been restored\nwith the same beam (reported in Tab. \\ref{tab:2345radiofinali} and\n\\ref{tab:1240radiofinali}) and corrected for the primary beam effects.\n\\begin{table}[h!]\n\\caption{ Abell 2345}\n\\label{tab:2345radiofinali}\n\\centering\n\\begin{tabular} {|l c c c c|} \n\\hline\\hline\nSource name & $\\nu$ & $\\theta$   & $\\sigma_{I}$ & $~~~$Fig. \\\\\n             &   MHz     & arcsec & mJy\/beam  & \\\\\n\\hline\nAbell 2345-1 HR &  1425      & 37 X 20    &  0.08 & \\\\\nAbell 2345-1 LR &  1425      & 50 X 38    &  0.09 & \\ref{fig:A2345_spixcut}, central panel\\\\\nAbell 2345-2 HR &  1425      & 37 X 20    &  0.09 & \\\\\nAbell 2345-2 LR &  1425      & 50 X 38    &  0.09 & \\ref{fig:A2345_spixcut}, central panel\\\\\nAbell 2345   HR &  325      & 37 X 20    &   1.7  & \\\\\nAbell 2345   LR &  325      & 50 X 38    &   2.0  & \\ref{fig:A2345_spixcut}, right panels\\\\\n\\hline\nAbell 2345    &  1490     &6X6         &   0.13  & \\ref{fig:A2345_otticoradio}, central panel\\\\\n\\hline\n\\multicolumn{5}{l}{\\scriptsize Col. 1: Source name; Col. 2: Observation frequency;\n}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col. 3: Restoring beam; Col. 4: RMS noise of the \nfinal images;}\\\\\n\\multicolumn{5}{l}{\\scriptsize  Col 5: Figure of merit.}\n\\end{tabular}\n\\end{table}\n\\begin{table}[h!]\n\\caption{ Abell 1240}\n\\label{tab:1240radiofinali}\n\\centering\n\\begin{tabular} {|l c c c c|} \n\\hline\\hline\nSource name & $\\nu$ & $\\theta$   & $\\sigma_{I}$ & $~~~$Fig. \\\\ \n             &   MHz     & arcsec & mJy\/beam &\\\\\n\\hline\nAbell 1240-1 HR &  1425      & 22 X 18    &  0.04&\\ref{fig:A1240_otticoradio.ps} \\\\\nAbell 1240-1 LR &  1425      & 42 X 33    &  0.04&\\ref{fig:A1240spix},central panel \\\\\nAbell 1240-2 HR &  1425      & 22 X 18    &  0.04&\\ref{fig:A1240_otticoradio.ps} \\\\\nAbell 1240-2 LR &  1425      & 42 X 33    &  0.05&\\ref{fig:A1240spix}, central panel \\\\\nAbell 1240   HR &  325      & 22 X 18    &  0.9 &\\\\\nAbell 1240   LR &  325      & 42 X 33    &  1.0& \\ref{fig:A1240spix}, left panels\\\\\n\\hline\n\\multicolumn{5}{l}{\\scriptsize Col. 1: Source name; Col. 2: Observation frequency;}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col. 3: Restoring beam; \n                    Col. 4: RMS noise of the final images;}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col. 5: Fig. of merit.}\n\\end{tabular}\n\\end{table}\n\\begin{table}[h!]\n\\caption{Total and polarization intensity radio images at 1425 MHz}\n\\label{tab:radiopol}\n\\centering\n\\begin{tabular}{|c  c c c  c| } \n\\hline\\hline \nSource name & $\\theta$ & $\\sigma_I$ & $\\sigma_{Q,U}$ & Fig. \\\\ \n           & arcsec & \\scriptsize(mJy\/beam) & \\scriptsize(mJy\/beam)&\\\\ \n\\hline \nAbell 2345-1 & 23 X 16 & 0.05 & 0.02 & \\ref{fig:A2345_pol},\nright panel\\\\ Abell 2345-2 & 23 X 16 & 0.07 & 0.02 &\n\\ref{fig:A2345_pol}, left panel\\\\ Abell 1240-1 & 18 X 17 & 0.04 & 0.02\n&\\ref{fig:A1240pol}, top panel\\\\ Abell 1240-2 & 18 X 17 & 0.04 & 0.01\n&\\ref{fig:A1240pol}, bottom panel \\\\ \\hline\n\\multicolumn{5}{l}{\\scriptsize Col. 1: Source name; Col. 2: Restoring\nbeam; }\\\\ \\multicolumn{5}{l}{ \\scriptsize Col. 4: RMS noise of the I\nimage; Col 5: RMS noise of the Q and U images}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col 6: Figure of merit.}\n\\end{tabular}\n\\end{table}\n\\subsection{Polarization intensity data reduction}\nObservations at 20 cm (1.425 GHz) include full polarization\ninformation. Polarization data observed with the D array are unusable\nbecause of bad quality of data of the polarization calibrator.  The\nabsolute polarization position angle has been calibrated by observing\n3C286 for both clusters in C configuration. The instrumental\npolarization of the antennas has been corrected using the source\n1156+314 for Abell 1240 and the source 2137-207 for Abell 2345.\\\\\nStokes parameters U and Q images have been obtained. We then derived\nthe Polarization intensity image ($P=\\sqrt{U^2+Q^2}$), the\nPolarization angle image ($\\Psi=\\frac{1}{2}arctan\\frac{U}{Q}$) and the\nFractional Polarization image ($FPOL=\\frac{P}{I}$), with I being the\ntotal intensity image. Further details are given in\nTab. \\ref{tab:radiopol}.\\\\\n\\begin{table*}\n\\caption{Abell 2345 and Abell 1240 properties}          \n\\label{tab:x}      \n\\centering          \n\\begin{tabular}{|c c c c c c c |}    \n\\hline\\hline       \nSource name &  RA          &   DEC       & z     &  scale    & F$_X$                   & L$_X$ \\\\\n            & (J2000)      & (J2000)     &       & (kpc\/$''$) & $10^{-12}$ erg\/s\/cm$^2$ & $10^{44}$ erg\/s \\\\\n\\hline\nAbell 2345  &  21 27 11.00 & -12 09 33.0 & 0.1765& 2.957     &  5.3                & 4.3\\\\\nAbell 1240  &  11 23  32.10 & 43 06 32   & 0.1590& 2.715     &  1.3               &  1.0\\\\\n\\hline\n\\multicolumn{7}{l}{\\scriptsize Col. 1: Source name; Col. 2, Col. 3: Cluster X-ray centre (RA,\nDEC); Col 4: Cluster redshift; Col 5: arcsec to kpc conversion scale;} \\\\\n\\multicolumn{7}{l}{\\scriptsize Col 6: Flux in the 0.1-\n2.4 keV band (Abell 2345) and in the 0.5-2 keV (Abell 1249); Col 7: X-ray cluster luminosity   }\\\\\n\\multicolumn{7}{l}{\\scriptsize in the 0.1-2.4 keV band (Abell 2345) and in the 0.5-2 keV (Abell 1240);}\\\\\n\\multicolumn{7}{l}{\\scriptsize Data from B\\\"ohringer et al. (2004) for Abell 2345 and from\nDavid et al. (1999) for  Abell 1240, corrected for the adopted cosmology.}\n\\end{tabular}\n\\end{table*}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=19.5cm]{comboA2345otticoradio.ps}\n\\caption{The cluster Abell 2345. In the center: DPOSSII optical\nemission (red band) in colors overlaid onto radio contours at 1.490\nGHz. First contours are $\\pm$ 0.4 mJy\/beam and are then spaced by a\nfactor 2. The beam in 6$''$$\\times$6$''$. Top inset shows the zoomed\nimages of the central sources: the central cD galaxy and two radio\ngalaxies are visible. Bottom inset shows the zoomed image of the\nsouthern radio source. Red Boxes mark the region of the relics,\ncompletely resolved in the high resolution image. In the Left and\nRight panels zoomed image of the red boxes is shown. Here colors\nrepresent the optical DPOSSII emission, while contours represents the\nrelic radio emission at the resolution of 23$''\\times$16$''$. The\nrelic A2345-1 is visible in the right panel, while A2345-2 is in the\nleft panel. Contours start at $\\pm$0.15mJy\/beam and are spaced by a\nfactor 2. Red arrows indicate the position of the discrete sources\nembedded in the relic emission.}\n\\label{fig:A2345_otticoradio}\n\\end{figure*}\n\\section{The Cluster Abell 2345}\nOptical information are available for this cluster, while little is\nknown about its X-ray emission. General data are reported in\nTab. \\ref{tab:x}.\\\\\nWeak gravitational lensing analysis has been performed by Dahle et al.\n(2002) and by Cypriano et al. (2004). Optical data cover the inner\npart of the cluster ($\\sim$3$'$$\\times$3$'$). They find that this cluster\nhas a well defined core dominated by a cD galaxy, and both the light\nand galaxy number density distribution have several peaks close to the\ncentral galaxy.  The authors derived that the projected mass\ndistribution has the most prominent peak displaced from the central cD\nby $\\sim$1.5$'$, although a secondary peak is closer to the central\ncD. No information about the possible presence of a cooling flow\nassociated with this galaxy is present in the literature. Dahle et\nal. (2002) conclude from their analysis that the cluster may be a\ndynamically young system. Cypriano et al. (2004) report the mass\ndistribution derived from weak lensing analysis and find that the best\nfit to their data is a singular isothermal ellipsoid with the main\naxis oriented in the E-W direction.\\\\\n\\smallskip\\\\ The radio emission of Abell 2345 is characterized by the\npresence of two relics visible in the NVSS (Giovannini et al. 1999).\\\\\nOur new VLA observations confirm the presence of two regions where\nnon-thermal emission is present at the cluster periphery, nearly\nsymmetrical with respect to the cluster center.  These new\nobservations together with the archive data, allow the study of the\ncluster radio emission in a wide range of resolutions going from\n$\\sim$6$''$ to $\\sim$50$''$. Therefore, it is possible to separate the\ncontribution of discrete sources whose emission is not related to the\nrelic's physical properties. In Fig. \\ref{fig:A2345_otticoradio} the\nradio emission of Abell 2345 at 6$''$ resolution is shown overlaid\nonto the optical emission (taken from the Digitalized Palomar Sky\nSurvey II, red band). Two central radio-tail sources are associated\nwith optical galaxies in the cluster center. The central cD is visible\nin the optical image. Relics are not visible in this image because of\nthe lack of short baselines. This confirms that the emission detected\nin lower resolution observations is indeed extended and it is not due\nto the blending of discrete sources. In the same figure we also report\nthe radio relic emission as detected by C array observations.  The\nwestern relic (Abell 2345-1) is located at $\\sim$ 1 Mpc from the\ncluster X-ray center while the eastern relic (Abell 2345-2) is $\\sim$\n890 kpc far from the cluster center (see\nTab. \\ref{tab:A2345_relics}).\\\\ There are several discrete sources in\nproximity of the western relic, A2345-1, visible in the 1.4 GHz image,\nthey are labeled with letters from A to F in the right panel of\nFig.\\ref{fig:A2345_otticoradio}. The sources A, C, D, E and F could be\nassociated with the optical galaxies visible in the DPOSSII image,\nwhereas B does not have any obvious optical identification.  Optical\nemission is present at 35$''$ in NE direction from the radio\npeak. This is larger then the error associated with the beam, that is\nonly 6$''\\times$6$''$ in the highest resolution image. We can then\nconclude that no optical counterpart of the B radio source is detected\nin the DPOSSII image. The sources D E and F are not visible in the 325\nMHz image (see Fig. \\ref{fig:A2345_spixcut}, top left panel). This is\nconsistent with a radio source having a spectral index\n$<1.2$\\footnote{The spectral index $\\alpha$ is derived according to\n$S_{\\nu} \\propto \\nu^{-\\alpha}$.}. There is only one discrete source\nin proximity of the relic A2345-2, labeled with G in the\nFig.\\ref{fig:A2345_otticoradio} without any obvious optical\nidentification. This source is also detected in the higher resolution\nimage. \\\\ The whole extension of the relics is properly revealed by LR\nimages (Fig. \\ref{fig:A2345_spixcut}).  The morphology of the relics\nis similar at 1.4 GHz and 325 MHz, although only the brightest regions\ncan be seen at 325 MHz due to the higher rms noise level of these\nobservations with respect to the 1.4 GHz ones.  The total flux of the\nrelics at the 2 frequencies, excluding the contribution of the\ndiscrete sources, are reported in Tab. \\ref{tab:A2345_relics}, where\nthe main physical parameters are summarized.\\\\ The relic A2345-1 shows\nan elongated shape at high resolution, while at lower resolution it\nshows a weaker wide emission extending in the West direction\ni.e. toward the cluster outskirts. We note that this circular\nfilamentary morphology is not seen in other double relic sources, as\ndiscussed in the introduction.\\\\\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=19cm]{comboA2345spix.ps}\n\\caption{Center: the cluster Abell 2345 radio emission at 1.4 GHz. The\nbeam is 50$''$$\\times$38$''$. Contours start at 3 $\\sigma$ (0.24\nmJy\/beam) and are then spaced by a factor 2. The cross marks the X-ray\ncluster center. Left: Colors represent the spectral index of the\nrelic A2345-1 (top) and A2345-2 (bottom) superimposed over the radio\nemission at 325 MHz (contours). The beam is 50$''$$\\times$38$''$,\ncontours start at 3$\\sigma$ (6 mJy\/beam) and are then spaced by a\nfactor 2. Right: Spectral index error image (colors) superimposed onto\nthe emission at 325 MHz (contours are as above). }\n\\label{fig:A2345_spixcut}\n\\end{figure*}\n\\begin{figure*}[t!]\n   \\label{fig:A23451spixprofile.ps}\n   \\centering\n   \\includegraphics[width=9cm]{comboA2345-1anuli.ps}\n    \\includegraphics[width=9cm]{comboA2345-2anuli.ps}\n  \\caption{Spectral index radial trend of A2345-1 (left) and A2345-2\n          (right), computed in shells of $\\sim 50''$ in width. It has\n          been computed excluding the contribution of the discrete\n          sources. Crosses refer to spectral index values computed in\n          shells where the mean brightness is $>3\\sigma$ at both 325\n          MHz and 1.4 GHz. Arrows are $3\\sigma$ upper limits on the\n          spectral index mean value (see text). The red cross refers\n          to the cluster X-ray center, the blue cross refers to the\n          center of the spherical shells. In the insets: displacement\n          of the shells over which the mean spectral index has been\n          computed. Circles refer to the discrete sources embedded in\n          the relic emission. The red cross refers to the cluster\n          X-ray center, the blue cross is the center of the spherical\n          shells. }\n   \\end{figure*}\n\\subsection{Spectral index analysis}\n\\label{A2345spectralindex}\nWe derived the spectral index image of the cluster's relics comparing\nthe LR images at 1.4 GHz and 325 MHz. The rms noise of the images are\nreported in Tab. \\ref{tab:2345radiofinali}. Spectral index and\nspectral index noise images are shown in\nFig. \\ref{fig:A2345_spixcut}. They have been obtained by considering\nonly pixels whose brightness is $> 3\\sigma$ at both frequencies. We\nnote that relics are more extended at 1.4 GHz than at 325 MHz. This\ncan be due to the different sensitivities at 1.4 GHz and 325\nMHz. Confusion and RFI strongly affect the low frequency image, where\nthe noise level is significantly higher than the thermal noise. A\nconsistent spectral index analysis has to consider the different\nextension at the two frequencies. In fact, as already pointed out by\nOrr\\'u et al. (2007) if we compute spectral index analysis considering\nonly regions that have a signal to noise ratio $>$ 3 at both\nfrequencies, we introduce a bias, since we are excluding a priori low\nspectral index regions, whose emission cannot be detected at 325\nMHz. For instance, the relic A2345-1 radio brightness at 1.4 GHz\ndecreases as the distance from the cluster center increases. The\nfainter region could be detected in the 325 MHz image only if its\nspectral index, $\\alpha$, were steeper than $\\sim$1.8.\\\\ In both of\nthe relics the spectral index is patchy. The spectral index rms is\n$\\sigma_{spix}\\sim$ 0.4 while the mean spectral index noise is $<$Spix\nNoise$>\\sim$ 0.1 for both relics. Thus, by comparing these two\nquantities we can conclude that spectral index features are\nstatistically significant.  \\\\ Our aim here is to investigate if there\nis a systematic variation of the relic spectral index with distance\nfrom the cluster center as found in other radio relics (e.g. 1253+275\nby Giovannini et al. 1991; Abell 3667 by R\\\"ottgering et al. 1997;\nAbell 2744 by Orr\\'u et al. 2007; Abell 2255 by Pizzo et al. 2008;\nAbell 521 by Giacintucci et al. 2008; ).\\\\ In order to properly\nobtain the radial trend of the spectral index, we integrated the radio\nbrightness at 325 MHz and 1.4 GHz in radial shells of $\\sim 50''$ in\nwidth wherever the 1.4 GHz brightness is $>3\\sigma$, and then we\ncomputed the value of the spectral index in each shell. We have\nexcluded the regions where discrete radio sources are embedded in the\nrelic emission (see insets in Fig. \\ref{fig:A23451spixprofile.ps}).\nThe shells have been centered in the extrapolated curvature center of\nthe relic A2345-2, that is 2.6$'$ South the cluster X-ray\ncenter. Shells are then parallel to the relics main axis. We computed\nthe integrated brightness in each shell at 20 and 90 cm , and\ncalculated the associated error as $\\sigma\\times\\sqrt{N_{beam}}$,\nwhere $\\sigma$ is the image rms noise, $N_{beam}$ is the number of\nbeams sampled in the shell.  In those shells where the brightness is\n$>3\\sigma$ in the 1.4 GHz image but $<3\\sigma$ in the 325 MHz image,\nonly upper limits on the mean spectral index can be derived. The\nspectral index profiles thus obtained are shown in\nFig. \\ref{fig:A23451spixprofile.ps}. These plots shows that the\nspectral index in the relic A2345-1 increases with distance from the\ncluster center, indicating a spectral steepening of the emitting\nparticles. The spectral index, in each shell, is rather high, going\nfrom $\\sim$1.4 in the inner rim to $\\sim$1.7 in the central rim of the\nrelic. The spectral index trend derived for the outer shells is\nconsistent with a further steepening.\\\\ The spectral index of the\nrelic A2345-2 shows instead a different trend, going from $\\sim$1.4 in\nthe inner shell to $\\sim$1.1 in the outer rim\n(Fig. \\ref{fig:A23451spixprofile.ps}).\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1.2\\columnwidth]{A2345g1.5Xradio.ps}\n\\caption{Abell 2345 X-ray emission (colors) in the energy band 0.1-2.4\nkeV from ROSAT PSPC observations. The image has been smoothed with a\nGaussian of $\\sigma\\sim$60$''$;\ncontours represent the radio image of the cluster at 1.4 GHz. The beam\nis 50$''$$\\times$38$''$. Contours are 0.24 mJy\/beam and are then spaced\nby a factor 2. Arrows mark the position of the X1 X2 and X3 regions. }\n\\label{fig:A2345X}\n\\end{figure*}\n\\subsection{Radio-X-ray comparison}\nNo X-ray studies are present in the literature for this cluster.\nX-ray observations in the energy band $0.1-2.4$ keV have been\nretrieved by the ROSAT data archive. The cluster is $\\sim$ 100$'$\noffset from the ROSAT pointing. Observations have been performed with\nthe ROSAT PSPC detector for a total exposure time of $\\sim$ 14\nksec. After background subtraction the event file has been divided by\nthe exposure map. We smoothed the resulting image with a Gaussian of\n$\\sigma=60''$. The resulting image is shown in\nFig. \\ref{fig:A2345X}.\\\\ The X-ray emission of this cluster is\nelongated in the NW-SE direction. Two bright regions are visible at\n$\\sim$ 10$'$ and 14$'$ in N-W direction from the cluster center (we\nrefer to them as X1 and X2 respectively). The galaxy J21263466-1207214\n(RA =21h26m34.6s, DEC= -12d07m22s, z=0.178221) is close to the first\none. Another bright region is present at $\\sim$ 4$'$ South from the\ncluster center (X3). \\\\ Data presented here allow an interesting\ncomparison among cluster emission at different wavelengths. We note\nthat mass distribution from weak lensing studies (Cypriano et\nal. 2004) is well represented by an ellipsoid with the major axis\ndirected in the E-W direction and relics are found perpendicular to\nthis axis. Consistently with the optical analysis, the X-ray emission\nis elongated in the NW-SE direction, indicating a possible merger\nalong that direction, and relics are displaced perpendicular to that\naxis. In Fig. \\ref{fig:A2345X} the X-ray emission is superimposed onto\nradio contours. A2345-2 is located at the edge of the X-ray emission,\nas found in relics of Abell 3667 and A3376. A2345-1, instead, is\nlocated between eastern edge of the cluster and the X1 region, 10$'$\nfrom Abell 2345 center, and its radio emission extends toward X1.\\\\\nFrom the same figure, in the X3 region a Narrow Angle Tail radio\ngalaxy is visible in radio images at every resolution (see\nFig.\\ref{fig:A2345_otticoradio} and \\ref{fig:A2345_spixcut}). Although\nredshift is not available for this radio source, its structure favors\na connection to the cluster and\/or to the close X-ray peak. One\npossibility is that these X-ray multiple features are galaxy clumps\ninteracting with Abell 2345. \\\\ A self consistent scenario arises from\nthis analysis, indicating that the cluster Abell 2345 could be\nundergoing multiple merger with X3 and X1 groups, and this could\nexplain the peculiar properties of A2345-1. More sensitive and\nresolved X-ray observations in conjunction with optical studies are\nrequired to shed light on the connection between the radio emission of\nA2345, X1 X2 and X3.\\\\\n\\subsection{Equipartition Magnetic field}\n\\label{secs:Equip}\nUnder the assumption that a radio source is in a minimum energy\nconditions, it is possible to derive an average estimate of the\nmagnetic field strength in the emitting volume (see e.g. Pacholczyk\n1970). We assume that the magnetic field and relativistic particles\nfill the whole volume of the relics, and that energy content in\nprotons and electrons is equal. We further assume that the volume of\nthe relics is well represented by an ellipsoid having the major and\nminor axis equal to the largest and smallest linear scale visible in\nour images; we estimated the third axis to be the mean between the\nmajor and minor one. The synchrotron luminosity is calculated from a\nlow frequency cut-off of 10 MHz to a high frequency cut-off of 10\nGHz. The emitting particle energy distribution is assumed to be a\npower law in this frequency range ($N(E)\\propto E^{-p}$), with\n$p=2\\alpha+1$. We used the mean value of $\\alpha=$1.5 and 1.3 for\nA2345-1 and A2345-2 respectively and found $B_{eq}\\sim$1.0 $\\mu$G in\nA2345-1 and 0.8 $\\mu$G in A2345-2. These values are consistent with\nequipartition magnetic field found in other relics.\\\\ It has been\npointed out by Brunetti et al. (1997) that synchrotron luminosity\nshould be calculated in a fixed range of electron energies rather than\nin a fixed range of radio frequencies (see also Beck \\& Krause\n2005). In fact, electron energy corresponding to a fixed frequency\ndepends on the magnetic field value, and thus the integration limits\nare variable in terms of the energy of the radiating particles. Given\nthe power law of the radiating particles and the high value of the\nradio spectral index, the lower limit is particularly relevant\nhere. We adopted a low-energy cut off of $\\gamma_{min}$=100 and\nassumed $\\gamma_{max}>>\\gamma_{min}$, obtaining $B'_{eq}\\sim$ 2.9\n$\\mu$G in A2345-1 and 2.2 $\\mu$G in A2345-2.\\\\ We derived the minimum\nnon thermal energy density in the relic sources from $B'_{eq}$\nobtaining $U_{min}\\sim$8.1 and 4.3 10$^{-13}$erg\/cm$^{-3}$ for A2345-1\nand A2345-2. The corresponding minimum non-thermal pressure is then\n$\\sim$5.0 and 2.7 10$^{-13}$erg\/cm$^{-3}$. \\\\ We are aware that\n  the extrapolation to low energies or frequencies could over estimate\n  the number of low energy electrons, leading to over estimate the\n  equipartition magnetic field if a spectral curvature is present.  We\n  note that a detailed study of the radio spectrum on a large\n  frequency range is available for three peripheral relics: the one in\n  Abell 786, in the Coma cluster (see Giovannini \\& Feretti, 2004 and\n  references therein) and in Abell 521 (Giacintucci et al. 2008). In\n  these relics a straight steep radio spectrum is observed. We also note that a\n  low frequency cut-off of 10 MHz and a magnetic field of\n  $\\sim$1$\\mu$G imply a low energy cut-off of\n  $\\gamma_{min}\\sim$1500. Thus, also if the spectrum of the emitting\n  particles is truncated at $\\gamma>$1500, both $B'_{eq}$ and $B_{eq}$\n  could over estimate the magnetic field strength. Future\n  low-frequency radio interferometers such as SKA and LWA will likely shed\n  light on this point. On the other hand, it is possible to derive an\n  independent estimate of the magnetic field from X-ray flux due to\n  inverse Compton scattering of CMB photons by relativistic electrons\n  in the relic source. These studies have been performed so far on a\n  scarce number of radio relics and have led to lower limits on the\n  magnetic field strength: $B>$0.8$\\mu$G in the relic 1140+203 of\n  Abell 1367 (Henriksen \\& Mushotzky 2001); $B>$1.05$\\mu$G in 1253+275\n  of the Coma cluster (Feretti \\& Neumann 2006; $B>$0.8$\\mu$G in\n  0917+75 in Rood27 cluster (Chen et al. 2008); and $B>$2.2$\\mu$G in\n  the relic 1401-33 in the Abell S753 cluster (Chen et al. 2008). In\n  these cases, the lower limits derived from IC arguments are\n  consistent with equipartition estimates, thus indicating that the\n  equipartition value could be used as a reasonable approximations of\n  the magnetic field strength in relics.\n\\begin{table*} [t]\n\\caption{ Abell 2345}\n\\label{tab:A2345_relics}\n\\centering\n\\begin{tabular} {|c  c c c c c c  |} \n\\hline\\hline\nSource name &  Proj. dist & LLS         & F$_{20 cm}$   &  F$_{90 cm}$ & B$_{eq}$ - B$'_{eq}$& $<\\alpha>$ \\\\ \n            &  kpc        & kpc         & mJy          &  mJy        & $\\mu$G     &             \\\\\n\\hline\nAbell 2345-1 &  340$''$=1000& 390$''$= 1150 &30.0$\\pm$0.5& 291$\\pm$ 4 &  1.0 -2.9  &    $1.5\\pm$0.1  \\\\\nAbell 2345-2 &  300$''$=890 & 510$''$= 1500 &29.0$\\pm$0.4& 188$\\pm$ 3 &  0.8 -2.2   &    $1.3\\pm$0.1  \\\\\n\n\\hline \\multicolumn{7}{l}{\\scriptsize Col. 1: Source name; Col. 2:\nprojected distance from the X-ray centroid; Col. 3: Largest linear\nscale measured on the 20 cm images.}\\\\ \n\\multicolumn{7}{l}{\\scriptsize Col. 4 and 5: Flux density at 20 and 90 cm; Col. 6: equipartition\nmagnetic field computed at fixed frequency - fixed energy }\\\\\n \\multicolumn{7}{l}{\\scriptsize (see Sec. \\ref{secs:Equip}); Col. 7: mean spectral index in\nregion where both 20 and 90 cm surface brightness is $>$ 3 $\\sigma$}\n\\end{tabular}\n\\end{table*}\n\\begin{figure*}[t!]\n   \\centering \\includegraphics[width=18cm]{combopol.ps}\n      \\caption{Abell 2345: Polarized emission of Abell 2345 at 1,4\n      GHz. In the central panel the polarized radio emission at 1.4 GHz\n      is shown. The restoring beam is $23''\\times 16''$. Left and\n      right panels: contours refer to the radio image Abell 2345-2 and\n      Abell 2345-1 (see Tab. \\ref{tab:radiopol} for further\n      details). Contours start from $3 \\sigma$ and are spaced by a\n      factor 2. E vectors are superimposed: line orientation indicates\n      the direction of the E field, while line length is proportional\n      to the polarization intensity (Left panel: 1$''$ corresponds to\n      5.5 $\\mu$Jy\/beam; Right panel: 1$''$ corresponds to 10\n      $\\mu$Jy\/beam)}\n         \\label{fig:A2345_pol}\n   \\end{figure*} \n\\subsection{Polarization analysis}\n\\label{sec:A2345pol}\nAnother important set of information about the magnetic field in the\nrelics can be derived through the study of polarized emission. As\npreviously mentioned, we could calibrate polarization only for\nobservations at 1.4 GHz with the C array.\\\\ In Fig. \\ref{fig:A2345_pol} the P radio image of\nthe cluster is shown. The noise achieved in the P, Q and U images\n(Tab. \\ref{tab:radiopol}) are lower than those obtained in total\nintensity image. In fact total intensity image are affected by\ndynamical range limitation due to the presence of powerful radio\nsources near our target. These sources are not strongly polarized, so\nthat P images are not affected by such limitation, and weaker\npolarized emission can be revealed. We note in fact that polarized\nradio emission of the relic A2345-2 reveals an arc-like structure that\nis more extended than in total intensity emission. The arc-like\nstructure of this relic indicates that the shock wave, possibly\nresponsible of the radio emission, has been originated $\\sim 2.6'$\nsouthern the present X-ray center.\\\\ The mean fractional polarization\nis $\\sim 22 \\%$ in A2345-2, reaching values up to 50\\% in the eastern\nregion. The relic A2345-1, shows a mean fractional polarization of\n$\\sim 14 \\%$ with higher polarized region ( $\\sim 60 $\\%) in the\nnorth-western part of the relic.  The amount of fractional\npolarization allows to estimate the level of order of the magnetic\nfield in the source. Following Burn (1966), if we assume that the\nmagnetic field is composed by an ordered component ${\\bf B_o}$ plus a\nrandom isotropic component represented by a Gaussian with variance\nequal to $2\/3 B_{r}^2$, it results\n\\begin{equation}\nP_{oss}=P_{int}\\frac{1}{1+(B_r^2\/B_o^2)}\n\\label{eq:pol}\n\\end{equation}\n where $P_{oss}$ is the observed fractional polarization, while\n$P_{intr}$ is given by $P_{intr}=\\frac{3\\delta+3}{3\\delta+7}$.  For\nthe relic A2345-1 we obtain that $B_r^2\/B_o^2 \\sim 4$, meaning that\nthe magnetic energy density in the random component is three times\nlarger than the one in the ordered component. For the relic A2345-2,\ninstead, we obtain that $B_r^2\/B_o^2 \\sim 2$. This indicates that the\nmagnetic field in the region of the relic A2345-2 has a higher degree\nof order. We also have to consider possible beam depolarization,\ninternal depolarization and ICM depolarization, so that what we can\nconclude from this analysis is $B_r^2\/B_o^2 < 4$ and $<2$ in A2345-1\nand A2345-2 respectively.\\\\ In A2345-1 the magnetic field is\nmainly aligned with the sharp edge of the radio emission, i.e. in the\nSW-NE direction. In the northern part of the relic the E vectors rotate\nand in the N-W part they are almost aligned toward the SW-NE\ndirection. In A2345-2 the E vectors are perpendicular to the relic\nmajor axis, following the arc-like structure that is marginally\nvisible in the total intensity image. \\\\\n\\subsection{Results on Abell 2345}\n\\label{Sec:A2345results}\nThe presented analysis confirms that non-thermal emission is\nassociated with the ICM of Abell 2345.\n\\begin{itemize}\n\\item {The properties of the western relic, A2345-1 are quite\n  peculiar.   We note indeed that its morphology is rather\n    circular and filamentary, its brightness distribution is higher in\n    the inner region of the relic and its spectral index steepens\n    toward the cluster periphery. Although the statistic is really\n    poor, these features have not been found in other double relics so\n    far. The level of polarization, the magnetic field direction\n    mainly aligned with the sharp edge of the radio emission, and the\n    value of the equipartition magnetic field are instead in agreement\n    with other observed relics.\\\\ Diffusive shock acceleration models\n    predict a steepening of the radio spectrum towards the cluster\n    center (e.g. Ensslin et al. 1998; Hoeft \\& Br{\\\"u}ggen 2007) as a\n    consequence of the electron energy losses after shock\n    acceleration. It is worth mentioning here that theoretical\n    predictions rely on some assumptions about the\n    shock symmetry and the magnetic field structure that could be not\n    representative of this specific cluster environment.  Moreover, if\n    the relic is not seen edge on, projection effects could further\n    complicate the observed radio emission. Taking all of these into\n    account, the observed spectral index trend of A2345-1 cannot be\n    used as an argument to exclude an outgoing shock wave. \\\\ We\n    however note that the position of A2345-1 is in between the main\n    cluster and the possibly merging group X1. Thus we suggest the\n    possibility that its radio properties could be affected by this\n    ongoing merger. In particular, if the relic is seen edge-on, and\n    if the magnetic field strength is almost uniform in the relic\n    region, the observed spectral index trend could be the sign of a\n    shock wave moving inward, toward the cluster center. It could\n    result from the interaction with X1. Detailed optical and X-ray\n    observations would be needed to shed light on this point.}\\\\\n\\item{The relic A2345-2 shows the classical feature of ``elongated\n  relic sources'' also found in double relics of Abell 3667 and Abell\n  3376, as well as in single relic sources as 1253+275 (Andernach et\n  al. 1984, Giovannini et al. 1991) and A521 (Ferrari 2003,\n  Giacintucci et al. 2008). It is located far from the cluster center,\n  its spectral index is steep with mean value $\\sim$ 1.3 and steepens\n  towards the cluster center, as expected by relic formation theories\n  if the relic is observed edge-on. The value of the\n  equipartition magnetic field, the direction of the E vectors and the\n  detected level of polarization are consistent with previous\n  observations of elongated relics and agree with expectations from\n  theoretical models as well.  The polarized emission image reveals\n  the arc-like structure of the relic A2345-2. If we assume that the\n  relic is originated by a spherical shock wave, we can infer the\n  propagation center of the shock by extrapolating the curvature\n  radius of the relic. It results that the propagation center is $\\sim\n  2.6'$, southern the present X-ray center of the cluster Abell 2345\n  (see Fig. \\ref{fig:A23451spixprofile.ps}).\\\\ This corresponds to a\n  physical distance of 450 kpc at this redshift.  From weak lensing\n  analysis the galaxy velocity dispersion in this cluster results\n  $\\sim$900 km\/s (Dahle et al. 2002; Cypriano et al. 2004). As we will\n  see in Sect.\\ref{sec:discussion}, the expected Mach number is of\n  about 2.2 for this relic. Since the galaxy velocity dispersion is\n  comparable to the sound speed in the ICM (see e.g. Sarazin 1988), a\n  Mach number 2.2 corresponds to a velocity of $\\sim$2000 km\/s.  The\n  relic A2345-2 is $\\sim$800 kpc far from the spherical-shock\n  center.  A shock wave with $M\\sim$2.2 travels this distance in\n    $\\sim$ 0.4 Gyr (if the shock speed remains constant). Thus the\n  merging between the two substructures should have occurred at $\\sim$\n  1200 km\/s to explain the shift of the X-ray center in this\n  scenario. This is a reasonable value for cluster merger\n  velocity.\\\\ Although a precise estimate should consider the amount\n  of energy injected in the ICM as the shock wave passes through it,\n  and despite the number of assumptions and approximations, we suggest\n  that the relic indicates the position of the merger center as it was\n  $\\sim$ 0.4 Gy ago. The time that the shock wave has taken to get the\n  present relic position is the time that the sub cluster has taken to\n  get the current X-ray center position.\\\\}\n\\end{itemize}\n\\begin{figure}[h!]\n\\centering\n   \\includegraphics[width=0.9\\columnwidth]{A1240otticoDPSSradio2.ps}\n      \\caption{Abell 1240. Colors: Optical emission from DPOSSII (red\n      band); Contours: radio emission at 1.4 GHz (HR image). Contours\n      start at $\\pm$3$\\sigma$ and are then spaced by 2. Red cross\n      signs the X-ray center, labels refer to the discrete sources\n      embedded in A1240-1. }\n         \\label{fig:A1240_otticoradio.ps}\n   \\end{figure}\n\n\\section{The Cluster Abell 1240}\nLittle is known in the literature about this cluster. It is a rich\ncluster classified as Bautz-Morgan type III. In Tab. \\ref{tab:x}\ngeneral data about this cluster are reported.\\\\ Kempner \\& Sarazin\n(2001) have revealed the presence of two roughly symmetric relics from\nthe Westerbork Northern Sky Survey (WENSS). From WENSS images relics\nare visible at 2 and 2.5 $\\sigma$ level. Our VLA observation confirm\nthe presence of two weak radio emitting regions in the cluster's\noutskirts. The radio image of the cluster is shown in\nFig. \\ref{fig:A1240_otticoradio.ps} (contours) overlaid into optical\nemission (from the DPOSSII, red band). The Northern relic (A1240-1) is\nlocated at $\\sim$ 270$''$ from the cluster X-ray center. This distance\ncorresponds to $\\sim$ 700 kpc at the cluster's redshift. This relic is\nmainly elongated in the E-W direction, and its radio brightness\ndecreases going from the western to the eastern  part of the relic (see\nFig. \\ref{fig:A1240spix}). At 325 MHz only the eastern brightest part\nis visible. This is likely due to the higher noise in the 325 MHz\nimage. In fact from the mean brightness of the weaker part of the\nrelic, we estimated that it should have a spectral index $>$3 to be\ndetected at 325 MHz. Three radio sources are embedded in the relic\nemission, they are labeled with A B and C in\nFig. \\ref{fig:A1240_otticoradio.ps}.  The sources A and B are not\ndetected in the 325 MHz observations. This is consistent with spectral\nindex values $<$1, as commonly found in radiogalaxies. A weak emission\nat 1.4 GHz links the A radio source at the relic (see\nFig. \\ref{fig:A1240spix}). \\\\\nThe southern relic (A1240-2) is located at\n$\\sim$ 400$''$ (1.1 Mpc) from the cluster X ray center. At 1.4 GHz it is\nelongated in the E-W direction extending $\\sim $ 480$''$. No discrete\nsources have been found embedded in the relic emission. Also in this\ncase at 325 MHz the relic's extension is reduced to $\\sim$ 350$''$ along\nthe main axis, and only the brightest regions are visible at 325\nMHz.\\\\ The relic's physical parameters are reported in\nTab. \\ref{tab:A1240_relics}. The quantity are computed excluding the\nregion where discrete sources (A,B and C) are present.\\\\\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1.99\\columnwidth]{comboA1240spix.ps}\n\\caption{Center: the cluster Abell 1240 radio emission at 1.4 GHz. The\nbeam is 42$''$$\\times$33$''$. Contours start at 3$\\sigma$ (0.13\nmJy\/beam) and are then spaced by a factor 2. The cross marks the\ncluster X-ray center. Left: colors represent the spectral index of the\nrelic A1240-1 (top) and A1240-2 (bottom) superimposed over the radio\nemission at 325 MHz (contours) The beam is 42$''$$\\times$33$''$, first\ncontours are 2 $\\sigma$ (2 mJy\/beam), 3$\\sigma$ and are then spaced by\na factor 2. Right: Spectral index error image (colors) superimposed\nonto the emission at 325 MHz (contours are as above).}\n\\label{fig:A1240spix}\n\\end{figure*}\n\\begin{figure*}[t!]\n    \\centering\n   \\includegraphics[width=9cm]{comboA1240-1anuli.ps}\n   \\includegraphics[width=9cm]{comboA1240-2anuli.ps}\n      \\caption{Spectral index radial trend of A1240-1 (left) and\n          A1240-2 (right), computed in shells of $\\sim 50''$ in\n          width. It has been computed excluding the contribution of\n          the discrete sources. Crosses refer to spectral index values\n          computed in shells where the mean brightness is $>3\\sigma$\n          at both 325 MHz and 1.4 GHz. Arrows are $3\\sigma$ upper\n          limits on the spectral index mean value (see text). In the\n          inset: displacement of the shells over which the mean\n          spectral index has been computed. Circles refer to the\n          discrete sources embedded in the relic emission. The red\n          cross refers to the cluster X-ray center, the blue cross is\n          the center of the spherical shells. }\n  \\label{fig:A1240spixprofile.ps}\n \\end{figure*}\n\n\\subsection{Spectral index analysis}\nWe report in Fig.\\ref{fig:A1240spix} the spectral index map and the\nspectral index map error for the relics of Abell 1240. They have been\nobtained considering only those pixels that have a brightness\n$>$2$\\sigma$ at both frequencies.\\\\ Fig. \\ref{fig:A1240spix} shows\nthat the spectral index image is patchy. The spectral index image rms,\n$\\sigma_{spix}$, is $\\sim$ 0.3 and 0.4 for A1240-1 and A1240-2\nrespectively, while the mean of the spectral index error image,\n$<$Spix Noise$>$ is $\\sim$ 0.2 for both of the relics. We can then\nconclude that features in A1240-2 are statistically significant, while\ngiven the small difference between $\\sigma_{spix}$ and $<$Spix\nNoise$>$ in A1240-1, we cannot exclude that local features are a noise\nartifact in this case. In the relic A1240-2 a gradient is visible\nalong the main axis of the relic, as has been found in Abell 2256 by\nClarke \\& Ensslin (2006).\\\\ In Fig. \\ref{fig:A1240spixprofile.ps} the\nradial spectral index trend is shown for A1240-1 and A1240-2\nrespectively. They have been obtained as described in\nSec. \\ref{A2345spectralindex}. Spherical shells are centered close to\nthe X-ray cluster center and they are parallel to the main axis of\nboth relics.\\\\ Despite the small extension of the relics at 325 MHZ,\nit is still possible to derive some important results on the spectral\nindex radial trends in these relics: in the relic A1240-1 the spectral\nindex is steeper in the inner part of the relic and flatter in the\nouter part, as found in A2345-2 and predicted by ``outgoing merger\nshock'' models if relics are seen edge-on (Roettiger et al. 1999;\nBagchi et al.  2006). The same trend is consistent with the spectral\nindex profile derived in A1240-2, although a firm conclusion cannot be\nderived from these data. We note in fact that errors and upper limit\nin the inner shell cannot exclude a constant spectral index or even an\nopposite trend.\\\\\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm]{A1240ROSAT60asecBand2+radio.ps}\n\\caption{Colors: Abell 1240X-ray emission in the energy band 0.5-2 keV\nfrom ROSAT PSPC observations. The image has been smoothed with a\nGaussian of $\\sigma\\sim$60$''$; contours represent the radio image of\nthe cluster at 1.4 GHz. The beam is 42$''$$\\times$33$''$. First contour is\n0.13 mJy\/beam, further contours are then spaced by a factor 2. }\n\\label{fig:A1240X}\n\\end{figure}  \n\\subsection{Radio-X-ray comparison}\n We retrieved from the ROSAT data archive X-ray observations in the\nenergy band $0.5-2$ keV. The cluster is $\\sim$ 28$'$ offset from the\ncenter of the ROSAT pointing. Observations have been performed with\nthe ROSAT PSPC detector for a total exposure time of $\\sim$ 12\nksec. After background subtraction the event file has been divided by\nthe exposure map. We smoothed the resulting image with a Gaussian of\n$\\sigma=60''$. The resulting image is shown in\nFig. \\ref{fig:A1240X}.\\\\In Fig. \\ref{fig:A1240X} the X-ray emission of\nthe cluster is superimposed onto radio contours. The X-ray emission of\nthis cluster is elongated in the S-N direction and shows a double\nX-ray morphology. As already stated by Kempner \\& Sarazin (2001) this\nmorphology is consistent with a slightly asymmetric merger. \\\\ Relics\nare located at the edge of the X-ray emission. Their emission shows\nthe characteristic elongated shape, and their main axis is\nperpendicular to the main axis of the X-ray emission, as found in\ndouble relics of Abell 3367 and Abell 3376.\\\\\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=3cm, angle=-90]{A1240-1pol.ps}\n\\centering\n\\includegraphics[width=3cm, angle=-90]{A1240-2pol.ps}\n\\caption{Top panel: A1240-1 radio emission at 1.4 GHz, lines represent\nthe E vectors. The line direction indicates the E vector direction and\nthe line length is proportional to the polarized flux intensity. 1$''$\ncorresponds to 3$\\mu$Jy\/beam. The beam is\n18$''$$\\times$18$''$. Contours start at 0.12 mJy\/beam and are then\nspaced by a factor 2. Bottom panel: A1240-2 radio emission at 1.4\nGHz. The line direction indicates the E vector direction and\nthe line length is proportional to the polarized flux intensity. 1$''$\ncorresponds to 2$\\mu$Jy\/beam. Contours are as above.}\n\\label{fig:A1240pol}\n\\end{figure}\n\\begin{table*} \n\\caption{ Abell 1240}\n\\label{tab:A1240_relics}\n\\centering\n\\begin{tabular} {|c  c c c c c c  |} \n\\hline\\hline\nSource name &  Proj. dist & LLS         & F$_{20 cm}$  &  F$_{90 cm}$ & B$_{eq}$ - B$'_{eq}$     & $<\\alpha>$ \\\\ \n            &  kpc        & kpc         & mJy         &  mJy        & $\\mu$G     &             \\\\\n\\hline\nAbell 1240-1 &  270$''$=700 & 240$''$= 650  &6.0$\\pm$0.2  & 21.0$\\pm$0.8& 1.0 -2.4   & 1.2  $\\pm$0.1  \\\\\nAbell 1240-2 &  400$''$=1100& 460$''$= 1250 &10.1$\\pm$0.4  & 28.5$\\pm$1.1& 1.0 -2.5  & 1.3   $\\pm$0.2  \\\\\n\\hline \\multicolumn{7}{l}{\\scriptsize Col. 1: Source name; Col. 2:\nprojected distance from the X-ray centroid; Col. 3: Largest linear\nscale measured on the 20 cm images.}\\\\ \\multicolumn{7}{l}{\\scriptsize\nCol. 4 and 5: Flux density at 20 and 90 cm; Col. 6: equipartition\nmagnetic field computed at fixed frequency - fixed energy (see Sec. \\ref{sec:A1240pol})\n}\\\\ \n\\multicolumn{7}{l}{\\scriptsize Col. 7: mean spectral index in\nregion where both 20 cm and 90cm  surface brightness is $>$ 3 $\\sigma$.}\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Equipartition magnetic field}\n\\label{sec:A1240pol}\nUnder the same assumptions explained in Sec. \\ref{sec:A2345pol} , we\ncalculated the equipartition magnetic field for the relics A1240-1 and\nA1240-2. Values obtained are reported in\nTab. \\ref{tab:A1240_relics}. We note that these values have been\ncomputed considering the brightness of those pixels for which we have\nwell constrained information about the spectral index value,\ni. e. those regions whose emission is detected at both\nfrequencies. Since the emission at 325 MHz is only detected in a small\nregion of the relics, while at 1.4 GHz relics are more extended, the\nequipartition estimates refer to the same small regions, and different\nestimates could be representative of the wider relic emission detected\nat 1.4 GHz.\\\\ We derived the minimum non thermal energy density in the\nrelic sources from $B'_{eq}$ obtaining $U_{min}\\sim$5.5\n10$^{-13}$erg\/cm$^{-3}$ for A1240-1 and A1240-2. The corresponding\nminimum non-thermal pressure in then $\\sim$3.4 and $\\sim$3.5\n10$^{-13}$erg\/cm$^{-3}$. The consistency between magnetic field\n  equipartition values and magnetic field lower limits derived by\n  X-ray emission in other few clusters (see discussion in\n  Sec. \\ref{secs:Equip}) indicates that equipartition magnetic field\n  can be used as a reasonable approximation of the magnetic field in relics.\n\\subsection{Polarization analysis}\nWe obtained the polarized intensity images for the relics as described\nin Sec. \\ref{sec:A2345pol}. In Tab. \\ref{tab:radiopol} the parameters\nrelative to the polarization images of the relics A1240-1 and A1240-2\nare reported.\\\\ In Fig. \\ref{fig:A1240pol} the polarized emission of\nthe two relics is shown. Observations performed with C array cannot\nreveal the weak extended emission, and thus only the most compact an\nbright regions are visible in this image. In these regions the\nmagnetic field is mainly aligned along the relic main axis in both of\nthe relics. This is consistent with what has been observed in the\nrelics of Abell 2345 and with what is expected from the models that\nexplain the origin of these sources (e. g. Ensslin et al. 1998;\nRoettiger et al. 1999). The mean fractional polarization of A1240-1 is\n26\\%, reaching values up to 70\\%. In the relic A1240-2 the mean\nfractional polarization is 29\\%, reaching values up to 70\\%. From\nEq. \\ref{eq:pol} we derive that $B_r^2\/B_o^2 \\sim$1.4 and 1.2\nrespectively. Because of possible beam depolarization, internal\ndepolarization and ICM depolarization, we conclude that\n$B_r^2\/B_o^2 <$1.4 and $<$1.2. This means that the magnetic energy\ndensity in the random and ordered component is similar.\n\\subsection{Results on Abell 1240 }\nOur observations confirm the presence of two relics in\nAbell 1240 with as steep spectral index values as $\\sim$1.2 and\n$\\sim$1.3. The spectral index trends derived for these relics indicate\na radial flattening toward the cluster outskirts. This\nis the trend predicted by ``outgoing merger shock'' models.\\\\ The\ndouble relics radio morphology and location are similar to the double\nrelics found in Abell 3667 and Abell 3376.\\\\ The polarization level is\nhigh in both of the relics, although we have to consider that our\npolarization observations lack the weak extended regions, that are\nprobably less polarized. The magnetic field estimate achieved under\nthe minimum total energy assumption reveals magnetic field of the\norder of $\\mu$G at the cluster periphery in the relic regions, ordered\non Mpc scale, indicating a magnetic field amplification and ordering.\n\\section{Discussion}\n\\label{sec:discussion}\nWe confirm the presence of double relics in the cluster Abell\n1240. Their symmetry and properties strongly suggest a common origin\nof A1240-1 and A1240-2.\\\\ In the cluster Abell 2345 we confirm the\nexistence of two relics. However, while A2345-2 is a classic extended\nperipheral relic source similar to 1253+275, in the Coma cluster (see\nGiovannini et al. 1991 and references therein), A2345-1 shows a more\ncomplex structure. We suggest that its properties could be due to its\npeculiar position in between the cluster Abell 2345 and the possibly\nmerging group X1, and thus affected by a more recent merger.\\\\ Several\nmodels have been proposed to explain the origin of radio relics. They\ncan be divided into 2 classes:\n\\begin{enumerate}\n\\item{Diffusive Shock Acceleration by Fermi-I process (Ensslin et\nal. 1998; Roettiger et al. 1999; Hoeft \\& Br{\\\"u}ggen 2007). }\n\\item{Re-acceleration of emitting particles due to adiabatic\ncompression of fossil radio plasma (Ensslin \\& Gopal-Krishna 2001). }\n\\end{enumerate} \nIn both of these models the presence of a shock within the gas is\nrequired.  The second one also requires the presence of a nearby radio\nsource to provide the fossil radio plasma which can be re-energized by\nthe shock wave. Simulations of cluster mergers show indeed that the\nmerging of two sub-clusters leads to the formation of shocks in the\ncluster outskirts (Ryu et al. 2003).\\\\ In favor of the second scenario\nthere is the observational evidence that relics resemble individual\nobjects and do not trace the entire shock front (Hoeft et al.\n2004). Moreover, when a radio ghost is passed by a shock wave with\ntypical velocity of 10$^3$ km\/s, it is adiabatically compressed\nbecause of the higher value of the sound speed in the radio ghost\n(Ensslin \\& Br\\\"uggen 2002). We have to remind, however, that the\nequation of state of the radio emitting plasma is still poorly known,\nand that if the radio plasma has a high mass load due to undetectable\ncool gas , it should get shocked (Ensslin \\& Gopal Krishna 2001).\\\\\nThe presence of double relics itself favors the first scenario,\nbecause of the low probability to find two symmetric regions with\nfossil radio plasma. \\\\ Several independent cosmological simulations\nhave identified two main categories of cosmological shocks:\\\\ (i)\n``accretion shocks'' resulting from accretion of cold gas onto already\nformed structure, characterized by high Mach numbers;\\\\ (ii) ``merging''\nor ``internal'' shocks due to merging of substructures such as galaxy\nclusters or groups, with moderate Mach numbers: $2\\le M \\le 4$ (see\nreview by Bykov et al. 2008 and references therein).\n\\subsection{Relics from merging shocks}\nThe presence of double relics is particularly interesting in this\nscenario since the shape, morphology and properties of these extended\nstructures strongly suggest the presence of shock waves propagating\nfrom the cluster center to the peripheral regions. Because of the\nshort radiative lifetime of relativistic electrons, radio emission is\nproduced close to the location of the shock waves. These models\npredict that the magnetic field is aligned with the shock front and\nthat the radio spectrum is flatter at the shock edge, where the radio\nbrightness is expected to decline sharply.  \\\\ The shock compression\nratio can be estimated from the radio spectral index $\\alpha$ (assuming an\nequilibrium electron population accelerated and cooled at the same\ntime, and assuming a polytropic index 5\/3, see Drury 1983), as\n\\begin{equation}\nR=\\frac{\\alpha+1}{\\alpha-0.5}.\n\\end{equation}\nThe pressure and temperature jumps across the shock can be estimated\n from the theory of shocks (Landau \\& Lifschitz 1966) as\n\\begin{equation}\n\\frac{P_2}{P_1}=\\frac{4R-1}{4-R}= \\frac{\\alpha+1.5}{\\alpha-1}; \\frac{T_2}{T_1}=\\frac{P_2}{RP_1}\n\\end{equation}\nhere and after the index 2 refers to down stream regions and 1 to\n up-stream regions i.e. regions inside and outside the cluster shock\n front. These parameters are reported in Tab. \\ref{tab:DSA}.\n\\begin{table*}\n\\caption{Predictions from the shock acceleration model}\n\\label{tab:DSA}\n\\centering\n\\begin{tabular}{|c c c c c c c |} \n\\hline\\hline\nRelic        &$\\alpha$ & M &  R    & $P_2\/P_1$    &  $T_2\/T_1$ & $(B_2\/B_1)_{isoP}$\\\\\n\\hline     \nAbell 2345-1 &1.5$\\pm$0.1 &2.8$\\pm$0.1&2.5$\\pm$0.2&6$\\pm$1  & 2.4$\\pm$0.4  &2.4$\\pm$0.2\\\\\nAbell 2345-2 &1.3$\\pm$0.1 &2.2$\\pm$0.1&2.9$\\pm$0.2&9$\\pm$3  & 3$\\pm$1      &3.0$\\pm$0.5\\\\\nAbell 1240-1 &1.2$\\pm$0.1 &3.3$\\pm$0.2&3.1$\\pm$0.3&14$\\pm$6 & 4$\\pm$2      &3.7$\\pm$0.8\\\\\nAbell 1240-2 &1.3$\\pm$0.2 &2.8$\\pm$0.3&2.9$\\pm$0.4&9$\\pm$3  & 3$\\pm$2      &3.0$\\pm$0.5\\\\\n\\hline\n\\multicolumn{7}{l}{\\scriptsize Col. 1: \nSource name; Col 2: spectral index value; Col. 3:  Mach number; Col 4: Shock compression ratio estimated   }\\\\\n\\multicolumn{7}{l}{\\scriptsize from the radio spectral index;  Col. 5, 6: \nPressure and temperature jump across the shock; Col. 7: Magnetic field  }\\\\\n\\multicolumn{7}{l}{\\scriptsize strength in the pre and post shock regions required to\nsupport the relic against the thermal pressure }\\\\\n\\end{tabular}\n\\end{table*}\nThe Mach number of the shock can be estimated from the radio spectral\nindex under some assumptions: if the emitting particles are linearly\naccelerated by shock, the spectral index of the particle energy\nspectrum p ($=2\\alpha+1$) is related to the Mach number $M$ of the\nshock through:\n\\begin{equation}\np=2\\frac{M^2+1}{M^2-1}+1\n\\end{equation}\nincluding the effect of particle aging (continuous injection and\nInverse Compton energy losses, see e.g. Sarazin 1999). Mach number\nvalues we obtained are reported in Tab. \\ref{tab:DSA}. These values\nare lower than Mach number expected for accretion shocks (e. g. Bykov\net al. 2008), and are instead consistent with those expected for\nweaker shocks due to merging of structures. \\\\ The spectral index\ntrend clearly detected in A2345-2 and in both relics of Abell 1240\nagrees with the predictions of this scenario. If relics are seen\nedge-on, the flattest region, in the outer part of the relics, would\ncorresponds to the current shock location, indicating shock waves\nmoving outward from the cluster center. As discussed\n  in Sec.  \\ref{Sec:A2345results}, A2345-1 shows a more complex radio\n  emission. It could be affected by a more recent merger with the X1\n  group. It could trace a merger shock moving inward to the cluster\n  center as a result of the Abell 2345 - X1 group interaction.\\\\\n\\subsubsection{Magnetic field and merging shocks}\n\\label{sec:Magnetic-Shock}\n The study of the magnetic field associated with the relics offers\n further opportunities to investigate the connection between relics\n and merger shock waves. First of all the presence of relics itself\n indicates the existence of significant magnetic field at the cluster\n periphery on the Mpc scale. Furthermore, the detected level of\n polarization shows that the magnetic field in these regions is rather\n ordered. \\\\ The effect of passage of a shock wave in the ICM\n   could be twofold: (i) order and compress a magnetic field that was\n   randomly oriented before the shock passage or (ii) compress a\n   magnetic field that was already ordered on the relic scale before\n   the shock passage . This depends on the turbulence development at\n   the cluster periphery, that could either give rise to a random\n   field in the cluster outskirts (case i) or not (case ii). Little is\n   known about this point from observational point of\n   view. Observational evidence from the gas pressure map of the Coma\n   cluster (Schuecker et al. 2004) indicates the relevance of chaotic\n   motions within the ICM.  Cosmological numerical simulations\n   (e.g. Bryan \\& Norman 1998; Sunyaev, Bryan \\& Norman 2003) suggest\n   that the level of ICM turbulence is larger at increasing radial\n   distances from the cluster center. If the simple\n   Kolmogorov's picture of incompressible fluid turbulence is assumed, this\n   implies a more developed turbulence in the outermost region (since\n   the decay time is L\/$\\sigma$, where L is the typical scale where\n   the bulk of turbulence is injected, and $\\sigma$ is the rms\n   velocity of turbulence). Recently Ryu et al. (2008) argued that\n   turbulence is likely well developed in clusters and\n   filaments, and not in more rarefied regions such as sheets and\n   voids. On the other hand Dolag et al.(2005) suggested that the bulk\n   of turbulence is injected in the core of galaxy clusters, thus\n   implying a more developed turbulence in the innermost regions,\n   compared to the outermost ones.  The main limitation of\n   cosmological simulations is the lack of resolutions in low density\n   environments, that makes it difficult to discriminate if the\n   turbulent cascade is developed in these regions. Moreover, details\n   of the conversion process of large\u2013scale velocity fields into MHD\n   modes is still poorly understood. Thus, from the theoretical point\n   of view the overall picture seems still uncertain.\\\\\n\n\n \n\n\n \n\n\n\n \n \n \n \n \n \nIn the case that the magnetic field in the cluster\noutskirts is randomly oriented before the shock passage (i. e. the\nturbulence is developed in the cluster outskirts) and that it has been\namplified and ordered, by the passage of the shock wave (case i\nabove), the observed ratio $B_r\/B_o$ derived by polarization analysis\n(Sec. \\ref{sec:A2345pol} and \\ref{sec:A1240pol}) could be used to\nestimate the magnetic field amplification due to the passage of the\nshock.\\\\\n Following Ensslin et al. (1998), if the relic is seen at some angle\n $\\delta>$0 between the line of sight and the normal of the shock\n front, the projected magnetic field should appear perpendicular to\n the line connecting the cluster center and the relic. This is indeed\n what polarization data presented here show. The magnetic field\n amplification, the observed integral polarization and the\n preferential direction of the field revealed by the E vectors\n orientation could be derived, provided that $\\delta$ and R, the shock\n compression factor, are known. Present data do not allow to infer the\n angle $\\delta$. Future X-ray and optical observations could\n reconstruct the merging geometry for these two clusters, as done,\n e.g.  in Abell 521 by Ferrari et al. (2003, 2006). Despite this, if\n relics are supported by magnetic pressure only, the upstream and\n downstream fields are related by $(B_2^2\/B_1^2)_{isoP}=P_2\/P_1$\n (``strong field'' case in Ensslin et al. 1998). This ratio can be\n compared to the ratio derived by the polarization properties of the\n relics, under the assumption that $B_2$ corresponds to the ordered\n component of the field and $B_1$ to the random one. In\n Tab. \\ref{tab:DSA} the $(B_2\/B_1)_{isoP}$ ratio is reported for the\n relics in Abell 2345 and Abell 1240. These values are comparable to\n the observed ratio $B_r\/B_o$ derived by polarization analysis\n (Sec. \\ref{sec:A2345pol} and \\ref{sec:A1240pol}).  \\\\ Another\n   indication of the magnetic field amplification in the relics may be\n   obtained by comparing the magnetic field in the relic with the\n   cluster magnetic field intensity expected at the relic\n   location. Relics are located at 700-1100 kpc from the cluster\n   center in Abell 2345 and Abell 1240. At these distances the cluster\n   magnetic field strength is expected to be of the order of\n   $\\sim$10$^{-1}\\mu$G (see e. g. Dolag et al. 2008; Ferrari et\n   al. 2008 and references therein).  Equipartition magnetic field\n   values are of the order of $\\mu$G (see Sec. \\ref{secs:Equip} and\n   \\ref{sec:A1240pol}), thus about 10 times higher. Despite the number\n   of uncertainties and assumptions related with the equipartition\n   estimate, this is consistent with the ratio $(B_2^2\/B_1^2)_{isoP}$\n   and $B_r\/B_o$.\\\\Even if no firm conclusion can be obtained by this\n   analysis, we can conclude at least that the drawn picture has no\n   inconsistencies with the presented observations.\n\n\n\n\n\n\\subsection{Relics from Adiabatic compression}\nAnother model to explain the origin of cluster radio relics has been\nproposed by Ensslin \\& Gopal-Krishna (2001). This idea has been\ninvestigated with the help of 3-dimensional Magneto Hydro Dynamical\nsimulations by Ensslin \\& Br\\\"uggen (2002) and in a more realistic\ncosmological environment by Hoeft et al. (2004). In this scenario\ncluster radio relics would originate by the compression of fossil\nradio plasma by shock wave occurring in the process of large scale\nstructure formation. The expected high sound velocity of that still\nrelativistic plasma should forbid the shock to penetrate into the\nradio plasma, so that shock acceleration is not expected in this\nmodel. The plasma gains energy adiabatically from the compression and\nthe magnetic field itself is amplified by such compression. If the\nelectron plasma is not older than 2 Gyr in the outskirts of a cluster,\nthey can emit radio wave again.  Simulations performed by Ensslin \\&\nBr\\\"uggen (2002) show that the radio morphology of the resulting radio\nrelic in the early stage after the shock passage is sheet-like. Then\nthe formation of a torus is expected when the post shock gas starts to\nexpand into the volume occupied by the radio plasma. Thus it is\nexpected in this scenario that some correlation should exist between\nthe morphology of the radio relic and its spectral index, that traces\nthe time passed after the shock wave has compressed and re-energized\nthe emitting particles. A2345-1 shows indeed a torus-like radio\nstructure and a spectral index higher than A2345-2, A1240-1 and\nA1240-2, that exhibit a sheet-like structure.  The simulations\nperformed by Ensslin \\& Br\\\"uggen (2002) indicate that the compression\nof the radio plasma by the shock can be estimated from a cluster radio\nrelic with a toroidal shape. Assuming the idealized case of a\ninitially spherical and finally toroidal radio cocoon, the compression\nfactor is given by:\n\\begin{equation}\n\\label{eq:plasma}\nR'=\\frac{2r_{max}^2}{3\\pi r_{min}^2}\n\\end{equation}\nwhere $r_{max}$ and $r_{min}$ refer to the outer and inner radius of\nthe torus. In the case of A2345-1 we assume that the observed torus\nlike structure can be described by taking $r_{max}\\sim$ the LLS of the\nrelic and $r_{min}$ the thickness of the filament in the N-E part of\nthe relic, as suggested by the same authors in the case of non perfect\ntoroidal filamentary relics. With $r_{max}\\sim$ 1 Mpc, $r_{min}\\sim$\n200 kpc it results $R'\\sim$ 5. This is higher than the value of\n  the maximum compression ratio for mono-atomic gas (that is 4); this\n  would indicate that the radio plasma has a different equation of\n  state. However no conclusion can be drawn since Eq. \\ref{eq:plasma}\n  is based on too simplistic assumptions, in particular a spherical\n  model for the compressed relic.\\\\\n\\section{Conclusions}\nWe have presented 1.4 GHz and 325 MHz observations of Abell 2345 and\nAbell 1240. The presence of double relics in these cluster had been\ninferred by Giovannini et al. (1999) for Abell 2345 and by Kempner \\&\nSarazin (2001) for Abell 1240 from NVSS and WENSS. We confirm the presence\nof two relics in each of these clusters.\nBy combining 1.4 GHz and 325 MHz observations we obtained the spectral\nindex image of the diffuse radio emission. The study of the polarized\nemission at 1.4 GHz has been presented as well.  The analysis of both\nthe spectral index distribution and the polarization properties of\nrelics allows to test several independent predictions of the relic\nformation models.\\\\ We report the summary of the results from the\npresented analysis:\\\\\n\\begin{enumerate}\n\\item{ {\\bf A2345}: two relics have been detected in the cluster\noutskirts at both 1.4 GHz and 325 MHz. They are not perfectly\nsymmetrical with respect to the cluster center; the normals to the\nrelic main axis form an angle of $\\sim$150$^{\\circ}$. A2345-2 is a\nclassical peripheral relic and A2345-1 is a peculiar relic with a\ntorus-like structure possibly related to a merging region.}\\\\\n\\item{{\\bf A1240}: relics are fainter than relics in A2345. Their\nextended emission is detected at 1.4 Ghz while only their brightest\npart are detected at 325 MHz. They are symmetrical with respect to the\ncluster center and the angle between their normals is\n$\\sim$180$^{\\circ}$ as found in the other known double relics: Abell 3667,\nAbell 3376 and RXCJ1314.4-2515.}\\\\\n\\item{ Relics are located at the edge of the X-ray emission of Abell 2345\nand Abell 1240. The X-ray emission of Abell 2345 shows multiple substructures\nthat could be galaxy groups interacting with A2345. Peculiar features\nof A2345-1 could arise from this multiple interaction, but only\ndetailed X-ray and optical analysis could shed light on this\npoint.}\\\\\n\\item{Relics in Abell 1240 are located perpendicular to the cluster main\naxis revealed by X-ray observations. The double X-ray morphology of\nthe cluster is typical of merging clusters. }\\\\\n\\item {The average spectral indexes are steep. We found 1.5 $\\pm$ 0.1\n  and 1.3 $\\pm$0.1 for A2345-1 and A2345-2 and 1.2$\\pm$ 0.1, 1.3$\\pm$\n  0.2 for A1240-1 and A1240-2.}\\\\\n\\item{The spectral index distribution in the relics is rather\nirregular and patchy, although this, a clear radial trend is present\nin the relics of these two clusters.  A2345-2 spectral index ranges\nfrom $\\sim$1.5 in the region closer to the cluster center to $\\sim$1.1\nin the outer rim. This trend is consistent with shock models\npredictions. The same trend is observed in both of Abell 1240\nrelics. A1240-1 spectral index ranges from $\\sim$1.1 to $\\sim$1.6\ngoing from the outer to the inner rim, A1240-2 spectral index is also\nconsistent with a similar trend (going from $\\alpha <$ 1.5 in the\ninner rim to $\\alpha\\sim$ 1.1 in the outer one).  An opposite trend is\ninstead detected in A2345-1. Spectral index values are lower in the\ninner rim ($\\sim$1.3) and increase toward the outer part of the relic\nreaching values $\\sim$1.7. This trend could be due to its peculiar\nposition between two merging clumps.}\\\\\n\\item{The magnetic field, as revealed by polarized emission is mainly\n  aligned with the relic main axis. In Abell 2345 the polarized\n  emission reveals the arc-like structure morphology of the relic\n  A2345-2. Under equipartition conditions, values of $\\sim$ 2.2 - 2.9 $\\mu$G are\n  derived. The field has been likely amplified,\n  consistently with shock models predictions.}\\\\\n\\end{enumerate}\n These results have been discussed in the framework of relic\n  formation models. The Mach numbers derived from the value of radio\n  spectral index disfavour the ``accretion shock'' scenario, since\n  they are too small. Outgoing merger shock waves, proposed to explain\n  double relic emission in Abell 3667 and A3376, could also work in\n  Abell 1240 and Abell 2345. For the latter cluster we suggest that\n  the peculiar emission of A2345-1 could be explained by a shock wave\n  moving inward, due to the interaction of the main cluster with the\n  X1 group.\\\\ The toroidal shape of A2345-1 could be produced by\n  adiabatic compression, however the available data and models do not\n  allow a conclusive comparison.\n \n\\begin{acknowledgements}\n    The authors grateful to Franco Vazza, Marica Branchesi and\n    Elisabetta Liuzzo for helpful discussions and to Klaus Dolag for\n    elucidating discussion on cluster magnetic fields. The authors\n    thank the anonymous referee for useful suggestions and\n    comments. NRAO is a facility of the National Science Foundation,\n    operated under cooperative agreement by Associated Universities,\n    Inc. This work was partly supported by the Italian Space Agency\n    (ASI), contract I\/088\/06\/0, by the Italian Ministry for University\n    and Research (MIUR) and by the Italian National Institute for\n    Astrophysics (INAF). This research has made use of the NASA\/IPAC\n    Extragalactic Data Base (NED) which is operated by the JPL,\n    California Institute of Technology, under contract with the\n    National Aeronautics and Space Administration.\n\\end{acknowledgements}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Conjectures}\nThe semantics of RDF makes it impossible to express contradictory points of view that act on the same data.\n\nEvery approach applied in the past has not solved the main issue, that is, being able to express different statements whose truth value is not known, or even in contrast with each other, without fully asserting them, therefore making them undoubtedly true.\n\nIn this paper we introduce and examine the characteristics of \\textit{conjectures}, a proposed extension to RDF 1.1 that by design allows the expressions of graphs whose truth value is unknown.\n\nThe approach revolves around two main concepts:\n\\begin{itemize}\n    \\item \\textit{conjecture}: a \\textit{concept whose truth value is not available} and its representation;\n    \\item \\textit{collapse to reality}: a mechanism to fully assert, when the time comes, the truth value of conjectures in their RDF-esque form.\n\\end{itemize}\n\nConjectures encapsulate plain RDF statements or other conjectures in specially marked named graphs.\nThey can be expressed in a strong form:\n\n\\begin{verbatim}\nCONJECTURE :deVereWroteHamlet {\n    :Hamlet dc:creator :EdwardDeVere .\n}\n\\end{verbatim}\n\nor, allowing us to be able to include them in plain RDF 1.1 datasets, in a weak form, where the predicate is expressed as a \\textit{conjectural predicate}:\n\n\n\\begin{verbatim}\n@prefix conj0001: <http:\/\/example.org\/exampleDoc#deVereWroteHamlet>\n\nGRAPH :deVereWroteHamlet {\n  :Hamlet conj0001:creator :EdwardDeVere .\n  conj0001:creator conj:isAConjecturalFormOf dc:creator . \n}\n:deVereWroteHamlet prov:wasAttributedTo :JThomasLooney .\n\n\\end{verbatim}\n\n\nAdditional statements adding information to the conjecture, but external to it, are the \\textit{grounds} or \\textit{ground statements}. One of them is the last triple in the last example. \n\nThis paper is organized as follows: \nin section 2 we show a number of fundamental concepts and results of the conceptual model of conjectures. \n\nIn section 3 we introduce the impact of conjectures onto the standard RDF Simple interpretation according to \\cite{HAYES04}. \n\nSections 4 through 7 describe the simple interpretation of the main concepts of Conjectures: conjectures of individual triples, collapse to reality of individual triples, conjectures and collapses of graphs, and conjectures involving blank nodes. \n\nIn section 8 we discuss the simple interpretation of the additional features of conjectures: nested conjectures, conjectural collapses, ad cascading collapses. \n\nIn section 9 we draw some conclusions about the formal model presented here.\n\n\\section{Set-theoretical formal model}\n\nAssume the disjoint sets $\\mathcal{I}$ (all IRIs), $\\mathcal{B}$ (blank nodes), and $\\mathcal{L}$ (literals). \nAn RDF triple is a tuple $(s, p, o) \\in \\mathcal{T} = (\\mathcal{I} \\cup \\mathcal{B}) \\times \\mathcal{I} \\times (\\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L})$. \n\nFor every RDF predicate $p$, let $\\mathcal{S}_{p} \\subseteq (\\mathcal{I} \\cup \\mathcal{B})$ be its domain and\n$\\mathcal{O}_{p} \\subseteq (\\mathcal{I} \\cup  \\mathcal{B} \\cup \\mathcal{L})$ its range. \n\nWe denote with \n\\begin{verbatim}\n    x { s p o } \n\\end{verbatim}\n\na triple $(s, p, o)$ that is referred to in the examples by\nthe name $x$.\n\n\\textbf{Conjecturing}: Conjecturing is the function $conj : \\mathcal{T} \\rightarrow \\mathcal{T}$ such that, for every RDF triple $t_{1} = (s_{1} , p, o_{1}), conj (t_{1}) = (s, p_{s,o} , o)$ where:\n\\begin{enumerate}\n    \\item Identity of subject: $s_{1} = s$.\n    \\item Identity of object: $o_{1} = o$.\n    \\item Disjointness: $\\forall s_{j}, o_{k}$ such that $(s_{j}, p_{s,o} , o_{k}) \\in \\mathcal{T}$, we have that $s = s_{j}$ and $o = o_{k}$ . \n\\end{enumerate}\n\n\n\\textbf{Conjectures, conjectural predicates, conjecturing triple}: given the triple $(s, p, o) \\in \\mathcal{T}$, its conjecture is the triple\n\n\\[conj((s, p, o)) = (s, p_{s,o}, o) .\\] \n\nConjectural predicates (or \\textit{weak predicates}) of predicate $p$ are all the predicates that are\nmembers of the set $\\mathcal{C}onj_{p}$ , such that:\n\n\\[ \\mathcal{C}onj_{p} = \\{cp \\in \\mathcal{I} | \\exists s \\in S_{p}, o \\in O_{p}, conj((s, p, o)) = (s, cp, o) \\}\\].\n\n\\textbf{Theorem 1}: every conjectural predicate is used in one triple only.\n\n\\textbf{Proof}: derives from item 3 of definition 1 (Disjointness):\nGiven two triples $(s_{1}, p_{s,o}, o_{1}) , (s_{2}, p_{s,o}, o_{2}) \\in \\mathcal{T}$.\n\nFor item 3 of definition 1 (Disjointness), we have that  $s_{1} = s_{2}$ and $o_{1} = o_{2}. \\forall (s, p, o) \\in \\mathcal{T}, \\exists! p_{s,o}$ such that $conj((s, p, o)) = (s, p_{s,o} , o)$.\n\n\\textbf{Corollary 1}: the function $conj$ is unique (barring predicate name changes).\n\n\\textbf{Proof}: derives immediately from Theorem 1.\n\n\\textbf{Conjectural Form}: predicate $q$ is said to be a \\textit{conjectural form} of $p$ if there exists a pair of subjects and objects $s, o$ such that $conj ((s, p, o)) = (s, q, o)$.\n\n\n\\textbf{Collapsing}: Collapsing is a function $collapses: \\mathcal{T} \\rightarrow \\mathcal{T}$ such that, for every RDF triple $t = (s, p, o)$,\n$collapses(s, p, o) = (s, p_{s,o} , o)$ iff $conj((s, p, o)) = (s, p_{s,o} , o)$, and is undefined otherwise.\n\n\\textbf{Collapsed predicate; collapse}: let $(s, p_{s,o}, o)$ be a conjecture. \nThe collapsed predicate of $p_{s,o}$ is the predicate $p$ such that \n\\[collapses (s, p, o) = (s, p_{s,o} , o)\\]. \nThe collapse is the triple \n\\[( (s, p, o), collapse, (s, p_{s,o}, o))\\].\n\n\\section{RDF Simple interpretation and Conjectures}\n\nIn RDF, a simple interpretation $I$ of a vocabulary $V$ consists of:\n\\begin{enumerate}\n\\item A non-empty set $IR$ of resources, called the domain or universe of $I$.\n\\item A set $IP$, called the set of properties of $I$.\n\\item A mapping $IEXT$ from $IP$ into the powerset of $IR \\times IR$ i.e. the set of sets\nof pairs $< x, y >$ with $x$ and $y$ in $IR$ .\n\\item A mapping $IS$ from IRIs into $IR$ - in order to map resources and properties\n\\item A partial mapping $IL$ from literals into $IR$ - in order to map literals\n\\end{enumerate}\n\n$IEXT(p)$ is the extension of $p$, that is the set of pairs that are the arguments for which the property $p$ is true.\n\nAccording to \\cite{HAYPSC14}, a semantic extension is a set of additional semantic assumptions that gives IRIs additional meanings on the basis of other specifications or conventions.\nWhen this happens, the semantic extension must conform to the  minimal truth conditions already enunciated, extending from them to accommodate the additional meanings.\n\nTherefore, we extend the RDF Simple Interpretation adding a new set of conjectural properties, $IPC$, disjoint from the set of properties $IP$, where the conjectural predicates are created on the fly.\n\nWe add a new mapping $IEXTC$ from $IPC$ to the Cartesian product $IR \\times IR$. $IEXTC(cp)$ identifies the pair for which the property $cp$ is true.\n\nBecause of the Disjointness property of the conjecturing function $conj$ seen in section 1, the pair satisfying the property $cp$ will always be unique.\n\nWe need to specify an additional mapping $CONJFORM$ from $IP$ into $IPC$ to map the conjectural forms of the properties. \n\nOur full simple interpretation of $I$ in RDF with Conjectures is:\n\n\\begin{enumerate}\n\\item A non-empty set $IR$ of resources, called the domain or universe of $I$.\n\\item A set $IP$, called the set of properties of $I$. \n\\item { \\color{blue} A set $IPC$, called the set of conjectural properties of $I$. $IPC \\cap IP = \\emptyset$ }\n\\item A mapping $IEXT$ from $IP$ into the powerset of $IR \\times IR$ i.e. the set of sets of pairs $< x, y >$ with $x, y \\in IR$ .\n\\item { \\color{blue} An injective mapping $IEXTC$ from $IPC$ into  $IR \\times IR$, in other words the set of pairs $< x, y >$ with $x, y \\in IR$. By definition of injective mapping, if $IEXTC(a)=IEXTC(b)$, then $a=b$, that is, any $cp \\in IPC$ uniquely applies to a specific pair $<x , y>$.}\n\\item { \\color{blue} A mapping $CONJFORM$ from $IP$ into $IPC$ in order to map the conjectural forms of the properties  }\n\\item A mapping $IS$ from IRIs into $IR$ - in order to map resources, properties and conjectural properties\n\\item A partial mapping $IL$ from literals into $IR$ - in order to map literals\n\\item {\\color{blue} A conjecture might be represented as a set of individual statements composing as a whole the conjecture. This set is a \\textit{conjectural graph}}\n\n\\end{enumerate}\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\textwidth]{Interpretation1}\n    \\caption{Schematics of the interpretation}\n    \\label{fig:Interpretation}\n\\end{figure}\n\n\n\\section{Conjectures}\n\\begin{itemize}\n\\item if $E$ is a literal then $I(E) = IL(E)$\n\\item if $E$ is an IRI then $I(E) = IS(E)$\n\\item if $E$ is a ground triple $s \\: p \\: o \\: .$ then\n    \\begin{itemize}\n        \\item $I(E) = true$ if $I(p) \\in IP$ and\n        \\item the pair $<I(s),I(o)> \\in IEXT(I(p))$\n        \\item otherwise $I(E) = false$.\n    \\end{itemize}\n\\item { \\color{blue} if $E$ is a Conjecture triple $s \\: cp \\: o \\:.$ then  }\n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n        \\item the pair $<I(s), I(o)> \\in IEXTC(I(cp))$\n        \\item $I(cp) \\in CONJFORM(I(p))$ for some $I(p) \\in IP$\n        \\item otherwise $I(E) = false$.\n    \\end{itemize}\n\\item if $E$ is a ground RDF graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.\n\\item { \\color{blue} if $E$ is a ground conjectural graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.}\n\\end{itemize}\n\n\nThe last clause captures the definition of conjectural graph seen before.\n\n\\subsection{Model}\nGiven a RDF graph $G$, we say that the interpretation $I$ is a model of the graph $G$ if all the triples of graph $G$ are satisfied, that is, if they are true according to rules of $I$.\nIn this case, we can say that Interpretation $I$ satisfies $G$.\n\nThe notion of model is at the basis of the (simple) entailment: \nfollowing standard terminology, we say that $I$ (simply) satisfies $G$ when $I(G)=true$, that $G$ is (simply) satisfiable when a simple interpretation exists which satisfies it, otherwise (simply) unsatisfiable, and that a graph $E$ simply entails a graph $G$ when every interpretation which satisfies $E$ also satisfies $G$. If two graphs $G$ and F each entail the other then they are logically equivalent. \\cite{HAYPSC14} \n\n\nIs our interpretation $I$ a model of this conjecture graph?\n\n\\begin{verbatim}\n:deVereWroteHamlet { \n   :Hamlet conj001:creator :EdwardDeVere .\n   conj001:creator conj:isAConjecturalFormOf dc:creator .\n}\n\\end{verbatim}\n\n\n\\begin{itemize}\n    \\item $IR=\\{dVWH, h, c, cc1, e, iacf\\}$\n    \\item $IP=\\{c, iacf\\}$\n    \\item $IPC=\\{cc1\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n    \\item $IS(:deVereWroteHamlet) \\rightarrow dVWH$\n    \\item $IS(:Hamlet)\\rightarrow h$\n    \\item $IS(conj001:creator)\\rightarrow cc1$\n    \\item $IS(:EdwardDeVere)\\rightarrow e$\n    \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n    \\item $IS(dc:creator)\\rightarrow c$\n    \\item $IEXT(c)=\\emptyset$\n    \\item $IEXT(iacf)=\\{<cc1,c>\\}$\n    \\item $IEXTC(cc1)=\\{<h,e>\\}$\n    \\item $CONJFORM(c) = cc1$\n\\end{itemize}\n\nIEXT(c) is the empty set. We are still dealing with a conjecture, there is no assertion regarding any \"real\" property (in this case $dc:creator$) yet.\n\nAll the clauses seem to hold. We can say that simple interpretation (s-interpretation) $I$ is a model of our graph.\n\n\\section{Collapse to reality} \\label{collapseToReality}\nAt a certain point, someone may want to consider our conjectures as true. \n\nIn this case we would need to specify a mechanism in order to express the statements in the conjecture in full force. \n\nThis mechanism is called \"collapse to reality\".\n\nIt consists of exactly two triples added to the base, where:\n\n\\begin{itemize}\n    \\item in the first new triple, the \"effective\" property is used instead of the corresponding conjectural property.\n    \\item the first new triple is then declared to \"collapse\" the conjecture .\n\\end {itemize}\n\nTwo new triples are added, nothing gets replaced or deleted. \nSo we can keep track of what's happening and all the relationships between the graphs.\n\nLet's reason on an example of a collapse:\n\n\\begin{verbatim}\n:attr1 { \n    :Hamlet conj003:creator :Shakespeare .\n    conj003:creator conj:isAConjecturalFormOf dc:creator.\n}\n\n:attr1Cot {\n    :Hamlet dc:creator :Shakespeare .\n}\n\n:attr1Cot  conj:collapses :attr1 .\n\n\\end{verbatim}\n\n$:attr1$ is collapsed by adding a triple $:attr1Cot$ where the conjectural predicate is replaced by the corresponding \"real\" predicate.\n\nThe final additional triple functions as the explication of the collapse. \n$:attr1Cot$ is declarad to collapse the conjecture $:attr1$ by means of the property \"conj:collapses\".\n\nFor the sake of clarity:\n\\begin{itemize}\n    \\item the conjecture triple to be collapsed will be the \"conjecture triple\";\n    \\item the new triple collapsing the conjecture will be the \"collapsing triple\";\n    \\item the last triple will be the \"collapse triple\".\n\\end{itemize}  \n\n\\subsection{Interpretation $[I + COLLAPSE]$} \\label{I+COLLAPSE}\n\nWe need to extend once more our Interpretation with Conjectures $I$ by adding the Collapse to Reality rules.\n\nWe define a new mapping $collapses$ from triples to triples that maps the relationship between the conjecture triple and its collapsing triple:\n\n$collapses(s,p,o)=(s,cp,o)$ iff $conj(s,p,o)=(s,cp,o)$.\n\nThe collapse collapses the conjecture triple if, and only if, the latter is the (unique) conjecture of the collapsing triple. \n\nWe can clearly see that the \"collapses triple\" is just the translation into RDF of the definition of collapse we have introduced in section 1, that is, the triple:\n\n\\[((s,p,o),\\quad collapses,\\quad (s,cp,o) )\\]  \n\nFrom a more formal point of view, given a conjectural triple $E$, we add the triple $E_{cot}$ as a collapsing triple $\\{s \\quad p \\quad o .\\}$.\n\nThe semantic conditions of the collapse to reality should be the following:\n\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E$ be a conjecture triple\n$\\\\ \\{s  \\quad  cp  \\quad o \\quad .\\}$\n\nlet $E_{cot}$ be a collapsing triple \n$\\\\ \\{ s \\quad p \\quad o \\quad . \\}$\n\nand finally let $E_{collapse}$ be the collapse triple\n$\\\\ \\{ E_{cot} \\quad \\mathit{collapses} \\quad E \\}$\n\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n        \\item $I(E_{cot}) = true$ if $I(p) \\in IP$ and\n        \\item $CONJFORM(I(p))=I(cp)$ and\n        \\item the pair $<I(s), I(o)> \\in IEXTC(I(cp))$ and\n        \\item the pair $<I(s), I(o)> \\in IEXT(I(p))$ and\n        \\item $I(E_{collapse}) = true$ if $collapses(I(E_{cot}))=I(E)$, that is:\n        $collapses(I(s),I(p),I(o))=(I(s),I(cp),I(o))$ \n        \\item otherwise $I(E) = false$ and $I(E_{cot}) = false$ and $I(E_{collapse})=false$.\n    \\end{itemize}\n\\end{itemize}\n\n\\subsection{Model}\n\nNow, let's check if our Interpretation  $[I + COLLAPSE]$  can be a model of our example graph.\n\\begin{itemize}\n    \\item $IR=\\{a1, h, cc3, s, iacf, c, a1cot, cl \\}$\n    \\item $IP=\\{iacf, c, cl\\}$\n    \\item $IPC=\\{cc3\\}$\n\\end{itemize}\nThe functions:\n\\begin{itemize}\n    \\item $IS(:attr1) \\rightarrow a1$\n    \\item $IS(:Hamlet)\\rightarrow h$\n    \\item $IS(conj003:creator)\\rightarrow cc3$\n    \\item $IS(:Shakespeare) \\rightarrow s$\n    \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n    \\item $IS(dc:creator)\\rightarrow c$\n    \\item $IS(:attr1Cot)\\rightarrow a1cot$\n    \\item $IS(conj:collapses)\\rightarrow cl$\n    \\item $IEXT(iacf)=\\{<cc3,c>\\}$\n    \\item $IEXT(c)=\\{<h,s>\\}$\n    \\item $IEXT(cl)=\\{<a1,a1cot>\\}$\n    \\item $IEXTC(cc3)=\\{<h,s>\\}$\n    \\item $CONJFORM(c) = cc3$\n\\end{itemize}\n\nLet's check our semantic conditions for the collapse to reality:\n\\begin{itemize}\n    \\item $cc3 \\in IPC$? Yes; \n    \\item $c \\in IP$? Yes;\n    \\item $CONJFORM(c)=cc3$? Yes;\n    \\item The pair $<h,s> \\in IEXTC(cc3)$? Yes;\n    \\item The pair $<h,s> \\in IEXTC(c)$? Yes;\n    \\item $collapses(h,c,s)=(h,cc3,s)$? Yes, because $CONJFORM(c)=cc3$, hence $conj(h,c,s)=(h,cc3,s)$\n\\end{itemize}\n\nAll of them seem to hold.\n\nHence, our interpretation $[I + COLLAPSE]$ is a model of our graph.\n\nAs we will see in section \\ref{CollapseGraphs}, the \\textit{collapsing triple} and the \\textit{collapse} triple can be safely merged into a \\textit{collapse graph}:\n\n\\begin{verbatim}\n:attr1 { \n    :Hamlet conj003:creator :Shakespeare .\n    conj003:creator conj:isAConjecturalFormOf dc:creator.\n}\n\n:collapseOfattr1 {\n    :Hamlet dc:creator :Shakespeare .\n    :collapseOfattr1 conj:collapses :attr1 .\n}\n\n\n\\end{verbatim}\n\n\n\\section{Conjectural Graphs and Collapse Graphs}\n\\subsection{Conjectural Graphs}\nA conjectural graph is a representation of a conjecture by means of a set of individual statements composing as a whole the said conjecture.\n\nA ground Conjectural Graph is a graph with no blank nodes.\n\nAll triples inside a Conjectural Graph must be either with a conjectural predicate used exactly once, or a \"conj:isAConjecturalFormOf\" triple connecting a conjectural predicate to an original one.\n\nThe semantic conditions for Conjectural Graphs are:\n\\begin{itemize}\n\\item if $E$ is a literal then $I(E) = IL(E)$\n\\item if $E$ is an IRI then $I(E) = IS(E)$\n\\item { \\color{blue} if $E$ is a Conjecture triple $s \\: cp \\: o \\:.$ then  }\n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n        \\item the pair $<I(s), I(o)> \\in IEXTC(I(cp))$\n        \\item $I(cp) \\in CONJFORM(I(p))$ for some $I(p) \\in IP$ and\n        \\item $E$ must be unique in the graph\n        \\item otherwise $I(E) = false$.\n    \\end{itemize}\n\\item { \\color{blue} if $E$ is a triple $cp conj:isAConjecturalFormOf p$ then }\n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n        \\item $I(p) \\in IP$ and\n        \\item $I(cp) \\in CONJFORM(I(p))$ and\n        \\item $I(conj:isAConjecturalFormOf) \\in IP$\n        \\item the pair $<I(cp), I(p)> \\in IEXT(I(conj:isAConjecturalFormOf))$\n        \\item otherwise $I(E) = false$\n    \\end{itemize}\n\\item { \\color{blue} if $E$ is a ground conjectural graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.}\n\\end{itemize}\n\n\n\\subsection{Collapse Graphs} \\label{CollapseGraphs}\n\nCollapse graphs are graphs that connect explicitly with a conjecture and make it possible to track which conjecture was collapsed.\n\nThey must be composed at least of:\n\\begin{itemize}\n    \\item the \"effective form\" of the conjecture to be collapsed \n    \\item a \"conj:collapse\" triple connecting the conjecture to the collapse\n\\end{itemize}\n\nThe semantic conditions are:\n\n\\begin{itemize}\n\\item let $E_{conj}$ be a conjecture triple $s \\: cp \\: o \\: .$\n\\item let $E_{collapse}$ be a collapse graph\n\\item if $E$ is a literal then $I(E) = IL(E)$\n\\item if $E$ is an IRI then $I(E) = IS(E)$\n\\item { \\color{blue} if $E$ is a triple $s \\: p \\: o \\:.$ then  }\n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(p) \\in IP$ and\n        \\item the pair $<I(s), I(o)> \\in IEXT(I(p))$\n        \\item $CONJFORM(I(p))=I(cp)$ where $cp$ is the conjectural predicate of the conjecture to be collapsed\n        \\item otherwise $I(E) = false$.\n    \\end{itemize}\n\\item { \\color{blue} if $E$ is a triple $E_{collapse} \\quad conj:collapses \\quad E_{conj}$ then }\n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(conj:collapses) \\in IP$ and\n        \\item the pair $<I(E_{collapse}), I(E_{conj})> \\in IEXT(I(conj:collapses))$\n        \\item otherwise $I(E) = false$\n    \\end{itemize}\n\\item { \\color{blue} if $E$ is a ground conjectural graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.}\n\\end{itemize}\n\n\n\n\n\n\\section{Blank Nodes}\nWith conjectures, we would want to be able to express even more \"uncertain\" concepts involving unnamed entities or unspecified values.\n\nIn RDF this is done through \\textit{blank nodes}, which indicate the existence of an entity without using a IRI to identify any particular one.\n\nIn this section we will use a simple sentence implying the reliance on a blank node, namely  \"Muammar al-Qaddafi claimed that the author of Othello was someone who was an Arab\" \\footnote{He really did it in December 1988 - see \"William Shakespeare's Othello\" by Jibesh Bhattacharyya, ISBN 9788126904747}. \n\nIn this case we don't know who someone is, and we can't identify it with any IRIs.\n\nNevertheless, the information we are conveying with our conjecture maintains some degree of meaningfulness, and it definitely is something we could reason upon.\n\nSticking to our examples' style, we could say it like this: \n\"it is conjectured that Othello was written by somebody and this somebody was an Arab. And this claim has been attributed to Muammar al-Qaddafi\".\n\nIn RDF:\n\\begin{verbatim}\n:ArabWroteOthello { \n  :Othello conj002:creator _:z .\n  conj002:creator conj:isAConjecturalFormOf dc:creator .\n  _:z dbpedia-owl:nationality :Arab .}\n\n:ArabWroteOthello prov:wasAttributedTo :MalQaddafi .\n\\end{verbatim}\n\nIt comes pretty natural to conceive the term \"somebody\" as a blank node.\n\n\\subsection{Interpretation $[I + A]$}\n\nIn order to deal with blank nodes, we need to add a new mapping $A$ from blank nodes into IR.\n\nTherefore, we extend our interpretation $I$:\n\\begin{itemize}\n\\item $[I + A](x)=I(x)$ for names\n\\item $[I + A](x)=A(x)$ if x is a blank node.\n\\end{itemize}\n\nWe add a couple of semantic conditions to our interpretation for blank nodes, one is the \"standard\" one for RDF graphs, the other one is for conjectures:\n\n\\begin{itemize}\n    \\item if $E$ is a RDF graph then $I(E) = true$ if $[I+A](E) = true$ for some mapping $A$ from the set of blank nodes in $E$ to $IR$, otherwise $I(E) =false$.\n    \\item { \\color{blue} if $E$ is a conjectural graph then $I(E) = true$ if $[I+A](E) = true$ for some mapping $A$ from the set of blank nodes in $E$ to $IR$, otherwise $I(E) =false$.}\n\\end{itemize}\n\n\\subsection{Model}\n\nIs our Interpretation $[I+A]$ with blank nodes a model of our example graph? \n\nLet's reason on $:ArabWroteOthello$ part only.\n\nBe $A$ our blank nodes mapping into $IR$: $\\_:z \\rightarrow zz$. \n\nOur interpretation $[I + A]$ will be: \n\n\\begin{itemize}\n    \\item $IR=\\{awo, o, c, cc2, iacf, n, a, zz \\}$\n    \\item $IP=\\{c, iacf, n\\}$\n    \\item $IPC=\\{cc2\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n    \\item $IS(:ArabWroteOthello) \\rightarrow awo$\n    \\item $IS(:Othello)\\rightarrow o$\n    \\item $IS(conj002:creator)\\rightarrow cc2$\n    \\item $IS(dc:creator)\\rightarrow c$\n    \\item $IS(dbpedia-owl:nationality) \\rightarrow n$\n    \\item $IS(:Arab)\\rightarrow a$\n    \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n    \\item $IEXT(c)=\\emptyset$\n    \\item $IEXT(iacf)=\\{<cc2,c>\\}$\n    \\item $CONJFORM(c) = cc2$\n    \\item $IEXT(n)= \\{<zz,a>\\}$\n    \\item $IEXTC(cc2)=\\{<o,zz>\\}$\n\\end{itemize}\n\nEven in this case $IEXT(c)$ is empty because there is no assertion regarding the property $dc:creator$ yet, so our conjecture is, well,  still a conjecture.\n\nThe last two functions hold because of the mapping $A$ of our blank node.\n\nAll the clauses are true. Our interpretation $I + A$ is a model of our graph.\n\nThis approach allows us to reason with the blank nodes in what-if scenarios, where we could define the $A$ mapping from blank nodes to specific resources of our choice, therefore exploring new relationships arising between the triples.\n\n\n\n\n\\section{Further applications of conjectures}\n\n\\subsection{Conjecture of a conjecture - nested conjectures}\n\nLet's imagine we want to express a conjecture on another conjecture.\nOf course, the process can involve as many conjectures over conjectures we want.\n\nSuch as:\n\n\\begin{verbatim}\n:conjecture01 {\n    :Hamlet conj004:creator :EdwardDeVere .\n    conj004:creator conj:isAConjecturalFormOf dc:creator .\n}\n\n:conjecture02 {\n :conjecture01 conj004:wasAttributedTo :JThomasLooney .\n conj004:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .\n}\n\n:conjecture03 {\n:conjecture02 conj004:wasInformedBy <https:\/\/bit.ly\/3wSH76A> .\nconj004:wasInformedBy conj:isAConjecturalFormOf prov:wasInformedBy .\n}\n\n:conjecture03 prov:wasAttributedTo :FabioVitali .    \n    \n\\end{verbatim}\nThey are three different conjectures, one becoming the subject of the other one:\n\\begin{itemize}\n    \\item the first one says that Hamlet is conjectured to have been written by Edward De Vere;\n    \\item the second one says that the previous conjecture could possibly have been made by J. Thomas Looney;\n    \\item The third one says that it might be that the second conjecture could have been brought to light by the resource with URL https:\/\/bit.ly\/3wSH76A.\n\\end{itemize}\nThe fourth triple says that the last conjecture has been attributed to Fabio Vitali.\n\nReading it (more or less) backwards:\nFabio Vitali has stated that the resource at https:\/\/bit.ly\/3wSH76A might have brought to light that J. Thomas Looney could have possibly said that Hamlet is conjectured to have been written by Edward De Vere.\n\nWe could imagine it as a stack, or a stair, where at level 0 we have the first conjecture, and then as we get on the higher steps, the conjectures we find are built with the conjecture at the level below as a subject (or object, or both):\n\\begin{verbatim}\nLevel 2:            E{3}={E{2}, cp3, o3}\nLevel 1:                  E{2}=(E{1}, cp2, o2}\nLevel 0:                        E{1}=(s1, cp1, o1)\n\\end{verbatim}\n \nDeveloping the Level 2 we see they are nested in each other:\n$E{3}=(E{1}, cp2, o2),cp3, o3)=(((s1, cp1, o1), cp2, o2), cp3, o3)$\n\n\\subsubsection{Interpretation $[I + NESTEDCONJ]$}\n\nIn order to delineate the rules for the conjectures of conjectures (or nested conjectures), we can reason with pairs of conjectures.\n\nAs stated before, the conjectures at levels $> 0$ could be based on lower-level conjectures as their subject or object, or both.\n\nFor the sake of conciseness, we momentarily limit ourselves to reason with the case of the lower-level conjectures as their subject only.\n\n\\begin{enumerate}\n    \\item if $E_{0}$ is a ground conjecture $(s_{0}, cp_{0}, o_{0})$, $E_{1}= (E_{0}, cp_{1}, o_{1})$\n    \\item if $E_{1}$ is a ground conjecture $(E_{0}, cp_{1}, o_{1})$, $E_{2}= (E{1}, cp_{2}, o_{2})$\n    [...]\n    \\item if $E_{k-1}$ is a \"higher level\" ground conjecture $(E_{k-2}, cp_{k-1}, o_{k-1})$, $E_{k}=(E_{k-1}, cp_{k}, o_{k})$  \n    \\item if $E_{k}$ is a \"higher level\" ground conjecture $(E_{k-1}, cp_{k}, o_{k})$, $E_{k+1}= (E_{k}, cp_{k+1}, o_{k+1})$  \n\\end{enumerate}\n\nWe should have the following cases:\n\\begin{itemize}\n    \\item base case:\n        \\begin{itemize}\n            \\item $E_{0}=(s_{0}, p_{0}, o_{0})$\n        \\end{itemize}\n    \\item 1st cases:\n          \\begin{itemize}\n              \\item $E_{1}=(E_{0}, cp_{1}, o_{1})$\n              \\item $E_{1}=(s_{1}, cp_{1}, E_{0})$\n          \\end{itemize}\n    \\item $k^{th}$ cases:\n        \\begin{itemize}\n            \\item $E_{k}=(E_{k-1)}, cp_{k}, o_{k})$\n            \\item $E_{k}=(s_{k}, cp_{k}, E_{k-1})$\n        \\end{itemize}\n\\end{itemize}\n\n\nLet's extend our interpretation $I$ adding new rules.\n\nThe extension will be subdivided into cases, depending on the type of conjectures to be evaluated.\n\nWe must also enforce an order on the operations: we start from the conjecture(s) at the \"lowest level\", that is, the one(s) not involving other conjectures, evaluate them and \"climb\" up the \"stair\".\n\nTherefore, the extension to the interpretation $I$ will be:\n\\newline\n\\textbf{Base case} - for the lowest level conjecture at level 0:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{0}$ be a conjecture triple\n$\\\\ \\{s_{0}  \\quad  cp_{0}  \\quad o_{0} \\quad .\\}$\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E_{0}) = true$ if $I(cp_{0}) \\in IPC$ and\n        \\item the pair $<I(s_{0}), I(o_{0})> \\in IEXTC(I(cp_{0}))$\n        \\item otherwise $I(E_{0}) = false$ \n    \\end{itemize}\n\\end{itemize}\n\nAs we \"climb\" up the \"stair\" and get to the conjecture $E_{k}$, we can assume that the conjecture $E_{k-1}$ has already been proved by the previous steps.\n\\newline\n\n\\textbf{$K^{th}$ Case S} - for conjectures at the first step and above having another conjecture as the subject:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{k-1}$ be a conjecture triple\n\nlet $E_{k}$ be a conjecture triple\n$\\\\ \\{ E_{k-1} \\quad cp_{k} \\quad o_{k}\\quad . \\}$\n\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E_{k}) = true$ if $I(E_{k-1}) = true$ and \n        \\item $I(cp_{k}) \\in IPC$ and\n        \\item the pair $<I(E_{k-1}),I(o_{k})> \\in IEXTC(I(cp_{k}))$\n        \\item otherwise $I(E_{k}) = false$\n    \\end{itemize}\n\\end{itemize}\n\n\n\\textbf{$K^{th}$ Case O} - for conjectures at the first step and above having another conjecture as the object:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{k-1}$ be a conjecture triple\n\n\nlet $E_{k}$ be a conjecture triple\n$\\\\ \\{ s_{k} \\quad cp_{k} \\quad E_{k-1} \\quad . \\}$\n\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E_{k}) = true$ if $I(E_{k-1}) = true$ and\n        \\item $I(cp_{k}) \\in IPC$ and\n        \\item the pair $<I(s_{k}), I(E_{k-1})> \\in IEXTC(I(cp_{k}))$\n        \\item otherwise $I(E_{k}) = false$\n    \\end{itemize}\n\\end{itemize}\n\n\n\\textbf{$K^{th}$ Case SO} - for conjectures at the first step and above having another conjecture as the subject and yet another one as the object:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{(k-1)a}$ be a conjecture triple\n\nlet $E_{(k-1)b}$ be a conjecture triple\n\nlet $E_{k}$ be a conjecture triple\n$\\\\ \\{ E_{(k-1)a} \\quad cp_{k} \\quad E_{(k-1)b} \\quad . \\}$\n\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E_{k}) = true$ if $I(E_{(k-1)a}) = true$ and\n        \\item $I(E_{(k-1)b}) = true$ and\n        \\item $I(cp_{k}) \\in IPC$ and\n        \\item the pair $<I(E_{(k-1)a}), I(E_{(k-1)b})> \\in IEXTC(I(cp_{k}))$\n        \\item otherwise $I(E_{k}) = false$\n    \\end{itemize}\n\\end{itemize}\n\n\n\\subsubsection{Model}\nIs our Interpretation $[I + NESTEDCONJ]$ a model of the nested conjectures example seen before?\n\nWe define the sets:\n\n\\begin{itemize}\n    \\item $IR=\\{c1, h, cc4, edv, iacf, c, c2, cwa4, jtl, pwa, c3, cwib4, http, pwib, fv\\}$\n    \\item $IP=\\{c, iacf, pwa, pwib\\}$\n    \\item $IPC=\\{cc4, cwa4, cwib4\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n    \\item $IS(:conjecture01) \\rightarrow c1$\n    \\item $IS(:Hamlet)\\rightarrow h$\n    \\item $IS(conj004:creator)\\rightarrow cc4$\n    \\item $IS(:EdwardDeVere)\\rightarrow edv$\n    \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n    \\item $IS(dc:creator)\\rightarrow c$\n    \\item $IS(:conjecture02) \\rightarrow c2$\n    \\item $IS(conj004:wasAttributedTo) \\rightarrow cwa4$\n    \\item $IS(:JThomasLooney)\\rightarrow jtl$\n    \\item $IS(prov:wasAttributedTo)\\rightarrow pwa$\n    \\item $IS(:conjecture03) \\rightarrow c3$\n    \\item $IS(conj004:wasInformedBy) \\rightarrow cwib4$\n    \\item $IL(<https:\/\/bit.ly\/3wSH76A>) \\rightarrow http$\n    \\item $IS(prov:wasInformedBy)\\rightarrow pwib$\n    \\item $IS(:FabioVitali) \\rightarrow fv$\n    \\item $IEXT(iacf)=\\{<cc4,c>,<cwa4,pwa>,<cwib4,pwib>\\}$\n    \\item $IEXT(c)=\\emptyset$\n    \\item $IEXT(pwa)=\\{<c3,fv>\\}$\n    \\item $IEXT(pwib)=\\emptyset$\n    \\item $IEXTC(cc4)=\\{<h,ev>\\}$\n    \\item $IEXTC(cwa4)=\\{<c1,jtl>\\}$\n    \\item $IEXTC(cwib4)=\\{<c2,http>\\}$\n    \\item $CONJFORM(c) = cc4$\n    \\item $CONJFORM(cwa4) = pwa$\n    \\item $CONJFORM(cwib4) = pwib$\n\\end{itemize}\n\nLet's check the validity of the rules of the new semantic extension.\n\nWe must start from the conjecture not depending on other conjecture, that is $:conjecture01$, and we use the Base Case:\n\n\\begin{itemize}\n        \\item Is $cc4 \\in IPC$? Yes\n        \\item Is the pair $<h, ev> \\in IEXTC(cc4)$ Yes\n    \\end{itemize}\n\nThe \"base\" conjecture seems to hold.\nWe can say that $c1 = true$.\n\\newline\n\nNow we \"climb the stair\" to $:conjecture02$. Since it is based on another conjecture ($:conjecture01$, already proved true) as its subject, we use \"$k^{th}$ Case S\" \n\n    \\begin{itemize}\n        \\item is $c1 = true$? Yes\n        \\item is $cwa4 \\in IPC$? Yes\n        \\item is the pair $<c1,jtl> \\in IEXTC(cwa4)$? Yes\n    \\end{itemize}\n\nThen we can say that $c2 = true$.\n\\newline\n\nLet's climb one step higher. $:conjecture03$ is based on $:conjecture02$ as its subject. We still use \"$k^{th}$ Case S\"\n\n    \\begin{itemize}\n        \\item is $c2 = true$? Yes\n        \\item is $cwib4 \\in IPC$? Yes\n        \\item is the pair $<c2,http> \\in IEXTC(cwib4)$? Yes\n    \\end{itemize}\n\n$\\rightarrow c3 = true$.\n\\newline\nThe last triple $:conjecture03 prov:wasAttributedTo :FabioVitali$ is satisfied by the rules of  the simple intepretation $I$\n\\newline\nEverything seems to hold.\n\nWe can say that our Interpretation $[I + NESTEDCONJ]$ satisfies all the triples of the graph.\n\n\n\n\\subsection{Conjecture of a collapse} \\label{ConjectureOfACollapse}\nLet us consider the following sentence: \n\n\\begin{quote}According to Encyclopaedia Britannica (\\url{https:\/\/bit.ly\/3qgakFT}), Samuel Johnson attributed Hamlet to William Shakespeare, and he was right in saying so.\\end{quote}\n\nThis sentence is more than a mere collapse: it is a conjecture by an article in Encyclopedia Britannica a) attributing to Samuel Johnson a conjecture (about the authorship of Hamlet), and b) expressing total confidence in such conjecture (i.e., collapsing it).  \n\nIts representation is:\n\n\\begin{verbatim}\n:attribution01 {\n  :Hamlet conj005:creator :WilliamShakespeare .\n  conj005:creator conj:isAConjecturalFormOf dc:creator .\n}\n\n:collapseOfAttribution01 {\n  :attribution01 conj005:wasAttributedTo :SamuelJohnson .\n  conj005:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .\n\n  :collapseOfAttribution01 conj005:collapses :attribution01 .\n  conj005:collapses conj:isAConjecturalFormOf conj:collapses .\n}\n\n:collapseOfattribution01 prov:wasInformedBy <https:\/\/bit.ly\/3qgakFT> .\n\\end{verbatim}\n\nIn this case we have a conjecture of a collapse that, if collapsing, as a side effect, triggers the collapse of another conjecture, therefore asserting it in full force.\nPlease note that, in its current and \"uncollapsed\" status, everything is conjectured.\n\nWe can see the familiar predicate $collapses$ in its conjectural form, not yet in force in this case.\nThis predicate, once met in its \"effective\" form, will trigger the collapse of its subject. \nWe will see it in the next section.\n\n\n\\subsection{Cascading collapses}\n\nCascading collapses are all the collapses that take place recursively when multiple and nested conjectures of collapses collapse.\n\nFor example, what if we wanted to state that the $collapseOfAttribution01$ is true? We would collapse it by adding the \"collapsing triple\" and the \"collapse triple\" as per the rules of Interpretation [I + COLLAPSE] seen before in \\ref{I+COLLAPSE}.\n\nFor the sake of simplicity, we resort to the notion of collapse graph (\\ref{CollapseGraphs}). According to that, we can safely group the added triples into the collapse graph $:collapseOfcollapseOfAttribution01$.\n\nNow that our conjecture of a collapse is declared true, therefore the collapse is true and effective, we have to enforce it, collapsing $:attribution01$\n\nGeneralising, once a conjecture of a collapse is declared true, we must enforce a new special rule for its collapse to reality:\n\nWhenever $collapses$ is met in its  effective form inside a collapse graph, its object, if it is a conjecture, must be collapsed.\n\nIf the object is a conjecture triple $s \\:cp \\:o$, we need to add the \\textit{collapsing triple} and the \\textit{collapse triple}.\n\nGetting back to our example, we can see that the \\textit{collapse triple} is already defined in its effective form, in fact we have that:\n\\begin{verbatim}\n    :collapseOfAttribution01 conj:collapses :attribution01 .\n\\end{verbatim}\n\nSo we just need to add the collapsing triple, stating the conjecture in its full force, in the graph where the $conj:collapses$ property is met in full force.\n\nThe result will be as follows:\n\n\\begin{verbatim}\n:attribution01 {\n  :Hamlet conj005:creator :WilliamShakespeare .\n  conj005:creator conj:isAConjecturalFormOf dc:creator .\n}\n\n:collapseOfAttribution01 {\n  :attribution01 conj005:wasAttributedTo :SamuelJohnson .\n  conj005:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .\n\n  :collapseOfAttribution01 conj005:collapses :attribution01  .\n  conj005:collapses conj:isAConjecturalFormOf conj:collapses .\n}\n:collapseOfattribution01 prov:wasInformedBy <https:\/\/bit.ly\/3qgakFT> .\n\n:collapseOfcollapseOfAttribution01  {\n    :attribution01 prov:wasAttributedTo :SamuelJohnson .\n    \n    :collapseOfAttribution01 conj:collapses :attribution01 .\n\n    :collapseOfcollapseOfAttribution01 conj:collapses :collapseOfAttribution01 . \n       .\n\n    :Hamlet dc:creator :WilliamShakespeare.\n}\n\n\\end{verbatim}\n\nA collapse graph is built for the first collapse of the conjecture of the collapse, and then, at the end, the final \"effective\" triple.\n\nWe need to extend the Interpretation [I + COLLAPSE] seen before in \\ref{I+COLLAPSE}.\n\nAfter we collapse the conjecture of the collapse following the rules in [I+COLLAPSE], we will find the property $collapses$ in its effective form inside the collapse graph.\n\nWe need to extend the notion of $collapses$ defining it from collapse graphs to conjectural graphs:\n$collapses(G)=(E)$ iff $\\forall (s,cp,o) \\in E, conj(s,p,o)=(s,cp,o) with (s,p,o) \\in G$.\n\nIf the object of the $collapses$ property is a triple, which can be considered as a special case of a graph, we will add its effective form to the conjectural graph $collapses$ is in. \n\nIf the object of $collapses$ property is a conjectural graph, we need to add the effective forms of all its conjectures.\n\nOne important thing to be aware of is that we disregard the subject of the $collapses$ property: the effective form of the conjecture(s) in the object(s) is added to the \\textit{current} collapse graph no matter their subject(s).\n\nIn the semantic conditions below we will simplify for clarity by adopting the notion of a generic graph as the theoretical subject of a $collapses$ property. The graphs will in fact be more than one, if the subjects of the conjectures which are objects of $collapses$ are distinct (as in the example above). \nAlso The conjecture graphs\/triples can be more than one.\n\nThe semantic conditions are:\n\n\\textbf{conjecture triple:}\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E$ be a conjecture triple\n$\\\\ \\{s  \\quad  cp  \\quad o \\quad .\\}$\n\nlet $E_{g}$ be a generic graph\n\nlet $E_{cg}$ be a collapse graph \n$\\\\ \\{ E_{g} \\quad \\mathit{conj:collapses} \\quad E \\\\\n       s \\quad p \\quad o . \\}$\n\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n        \\item $I(E_{cg}) = true$ if $I(p) \\in IP$ and\n        \\item $CONJFORM(I(p))=I(cp)$ and\n        \\item the pair $<I(s),I(o)> \\in IEXTC(I(cp))$ and\n        \\item the pair $<I(s),I(o)> \\in IEXT(I(p))$ and\n        \\item $I(conj:collapses) \\in IP$ and\n        \\item the pair $<I(E_{g}),I(E> \\in IEXT(I(conj:collapses))$\n        \\item otherwise $I(E) = false$ and $I(E_{cg})$ is false.\n    \\end{itemize}\n\\end{itemize}\n\n\\textbf{conjectural graph:}\nIf we are dealing with a conjectural graph as the object of $conj:collapses$:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E$ be a conjectural graph\n$\\\\ \\{s_{0}  \\quad  cp_{0}  \\quad o_{0} \\quad . \\\\\n      {[...]} \\\\\n      s_{k}  \\quad  cp_{k}  \\quad o_{k} \\quad . \\}$\n\nlet $E_{g}$ be a generic graph\n\nlet $E_{cg}$ be a collapse graph \n$\\\\ \\{ s_{0} \\quad p_{0} \\quad o_{0} . \\\\\n       {[...]} \\\\\n       s_{k}  \\quad  p_{k}  \\quad o_{k} \\quad . \\\\\n       E_{g} \\quad \\mathit{conj:collapses} \\quad E\\}$\n\nthen  } \n    \\begin{itemize}\n    \\color{blue}\n        \\item $I(E) = true$ if $I(cp_{k}) \\in IPC \\forall cp_{k} in E$ and\n        \\item $I(E_{cg}) = true$ if $I(p_{k}) \\in IP \\forall p_{k} in E_{cg}$ and\n        \\item $CONJFORM(I(p_{k}))=I(cp_{k}) \\forall p_{k} in E$ and\n        \\item any pair $<I(s_{k}),I(o_{k})> \\in IEXTC(I(cp_{k}))$ and\n        \\item any pair $<I(s_{k}),I(o_{k})> \\in IEXT(I(p_{k}))$ and\n        \\item $I(conj:collapses) \\in IP$ and\n        \\item the pair $<I(E_{g}),I(E)> \\in IEXT(I(conj:collapses))$\n        \\item otherwise $I(E) = false$ and $I(E_{cg})$ is false.\n    \\end{itemize}\n\\end{itemize}\n\n\\subsubsection{Model}\n\nLet's check once again if the interpretation can be a model for our last example.\n\nWe define the sets:\n\n\\begin{itemize}\n    \\item $IR=\\{a1, h, cc5, ws, iacf, c, coa1, c5wat, sj, pwat, c5cl, cl, pwib, url, ccoa1  \\}$\n    \\item $IP=\\{c, iacf, pwat, pwib, cl\\}$\n    \\item $IPC=\\{cc5, c5wat, c5cl\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n    \\item $IS(:attribution01) \\rightarrow a1$\n    \\item $IS(:Hamlet)\\rightarrow h$\n    \\item $IS(conj005:creator)\\rightarrow cc5$\n    \\item $IS(:WilliamShakespeare)\\rightarrow ws$\n    \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n    \\item $IS(dc:creator)\\rightarrow c$\n    \\item $IS(:collapseOfAttribution01) \\rightarrow coa1$\n    \\item $IS(conj005:wasAttributedTo) \\rightarrow c5wat$\n    \\item $IS(:SamuelJohnson)\\rightarrow sj$\n    \\item $IS(prov:wasAttributedTo)\\rightarrow pwat$\n    \\item $IS(conj005:collapses)\\rightarrow c5cl$\n    \\item $IS(conj:collapses)\\rightarrow cl$\n    \\item $IS(prov:wasInformedBy)\\rightarrow pwib$\n    \\item $IL(<https:\/\/bit.ly\/3qgakFT>)\\rightarrow url$\n    \\item $IS(:collapseOfcollapseOfAttribution01) \\rightarrow ccoa1$\n    \\item $IEXT(c)=\\{<h, ws>\\}$\n    \\item $IEXT(iacf)=\\{<cc5,c>, <c5wat,pwat>, <c5cl,cl>\\}$\n    \\item $IEXT(pwat)=\\{<a1, sj>\\}$\n    \\item $IEXT(cl)=\\{<a1, coa1>, <coa1, ccoa1>\\}$\n    \\item $IEXT(pwib)=\\{<coa1, url>\\}$\n    \\item $IEXTC(cc5)=\\{<h,ws>\\}$\n    \\item $IEXTC(c5wat)=\\{<a1,sj>\\}$\n    \\item $IEXTC(c5cl)=\\{<a1,coa1>\\}$\n    \\item $CONJFORM(c) = cc5$\n    \\item $CONJFORM(pwat) = c5wat$\n    \\item $CONJFORM(cl) = c5cl$\n\\end{itemize}\n\n\nLet's check now the rules. \nWe focus the check on the cascading collapses only. \n\nWe need to proceed in steps. The first $conj:collapses$ in our final collapse graph has $:attribution01$ as its object.\nWe can consider $:attribution01$ as a single triple, as it contains only one conjecture:\n\n\\begin{itemize}\n\\color{blue}\n    \\item is $cc5 \\in IPC$? Yes;\n    \\item is $c \\in IP$? Yes;\n    \\item is $CONJFORM(c)=cc5$? Yes;\n    \\item is the pair $<h,ws> \\in IEXTC(cc5)$? Yes;\n    \\item is the pair $<h,ws> \\in IEXT(c)$? Yes;\n    \\item is $cl \\in IP$? Yes;\n    \\item the pair $<a1,coa1> \\in IEXT(cl)$?Yes.\n\\end{itemize}\n\nIt seems to hold.\n\nLet's get to the second $con:collapses$ of our final collapse graph. It has $:collapseOfAttribution01$ as its object.\nIt is a graph. Let's apply rules for the graphs:\n\n\n    \\begin{itemize}\n    \\color{blue}\n        \\item $c5wat, c5cl \\in IPC$? Yes;\n        \\item $pwat, cl \\in IP$? Yes;\n        \\item $CONJFORM(c5wat)=pwat; CONJFORM(c5cl)=cl?$ Yes;\n        \\item $<a1,sj> \\in IEXTC(c5wat)$? Yes; $<a1, coa1> \\in IEXTC(c5cl)$? Yes;\n        \\item $<a1,sj)> \\in IEXT(pwat)$? Yes; $<a1, coa1> \\in IEXT(cl)$? Yes;\n        \\item $cl \\in IP$? Yes;\n        \\item the pair $<a1,coa1)> \\in IEXT(cl)$? Yes\n    \\end{itemize}\n\n\nIt can be easily verified that all the other triples of the example can be satisfied by the interpretation [I + COLLAPSE].\n\nOur interpretation is a model of the graph.\n\n\\section{Conclusions}\nIn this paper we have attempted to build a formal representation of the semantics of the \\textit{conjectures}.\n\nWe have started with the formal model and extended the simple interpretation and model of RDF 1.1.\n\nThen we have explored the key features of the \\textit{conjectures} and verified that they fit in the semantics, expanding it when needed.\n\nThe \\textit{conjectures} allow to express concepts whose truth value is unknown, without asserting it, something that is currently missing in RDF.\n\nWith this work we have demonstrated that, with a somewhat limited extension to its model, it is possibile to add this powerful feature to RDF 1.1.\n\n\\printbibliography[\ntitle={Bibliography}\n]\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{{INTRODUCTION}}\n\\label{introduction}\n\nProcessing the enormous volume of data sets generated by the social networks such as Facebook\n\\cite{facebook}\nand Twitter\n\\cite{twitter}\n, financial institutions, health-care industry, etc. has become a major motivations for data-parallel processing. In data-parallel processing applications, an incoming task needs a specific piece of data which is physically replicated on different servers. The task receives the service from a server which is close to the physical location of the stored data faster than another server far from the location where the data is stored. For instance, for a task, the speed of receiving the service from a server which has the data stored on its memory or local disk is much faster than receiving the service from a server which does not have the data. That forces the server to fetch the data from another server from the same rack, or even other remote racks. Unless the speed of data center networks have been increased, the differences between service time is still obviously large \\cite{intro1}, \\cite{intro2}, \\cite{intro3}, \\cite{nd1}. While assigning tasks, it is a critical consideration to schedule a task on a local server \\cite{nd1}, \\cite{nd2}, \\cite{nd3}, and \\cite{nd4}. Scheduling in this setting is called the near-data scheduling problem, or scheduling with data locality. As a result, scheduling and assigning tasks to more suitable servers makes the system stable in the capacity region of the system, and also reduces the mean delay experienced by all tasks in average sense. Therefore, to evaluate different algorithms for scheduling tasks to different servers, two optimality criteria are defined as follows:\n\\begin{itemize}\n\\item Throughput Optimality: A throughput optimal algorithm stabilizes the system in the whole capacity region. That is, the algorithm is robust to arrival rate changes, as long as the arrival rate is strictly in the capacity region of the system.\n\\item Heavy-traffic Optimality: A heavy-traffic optimal algorithm minimizes the mean task completion time as the arrival rate vector approaches the boundary of the capacity region (note that task completion time consists of both waiting time of the task to be assigned to a server and the service time). As a result, a heavy-traffic optimal algorithm assigns tasks to servers efficiently when the system is in the peak loads close to the capacity region boundary. This not only stabilizes the system, but also minimizes the mean delay in stressed load conditions.\\\\\n\\end{itemize}\n\nAlthough there are various heuristic algorithms taking the near-data scheduling into consideration for data centers with multiple levels of data locality \\cite{heuristic1}, \\cite{nd1}, \\cite{heuristic2}, their throughput and heavy-traffic optimality properties are not studied. In this paper, we discuss different real-time scheduling algorithms for systems with multiple levels of data locality with theoretical guaranties for throughput and heavy-traffic optimality. We also compare those algorithms against each other and evaluate them for mean task completion time in the capacity region, including both in high load and low load.\n\nIn the following, we first explain the common data center structures and problem definition. In Section \\ref{affinity}, we discuss prime algorithms such as Fluid Model Planning and Generalized c$\\mu$-Rule. We then discuss the problem for two and then three levels of data locality, in Section \\ref{twolevel} and Section \\ref{threelevel}, respectively. The paper is concluded in Section \\ref{conclusion}.\n\n\\section{{SYSTEM MODEL}}\n\\label{systemmodel}\nConsider the system consists of $M$ servers indexed by $m \\in \\{1, 2, \\dots, M \\} \\overset{\\Delta}{=} \\mathcal{M}$. Each server belongs to a specific rack denoted by $k \\in \\{1, 2, \\dots, K \\} \\overset{\\Delta}{=} \\mathcal{K}$. Without loss of generality, let's assume that the servers in a rack are indexed sequentially. That is, servers $\\{ 1, 2, \\dots, i \\}$ are in the first rack, servers $\\{ i + 1, i + 2, \\dots, j \\}$ are in the second rack, and so on. Let the rack for the server $m$ be $K(m)$. Each data chunk is stored on a set of servers denoted by $\\bar{L}$. In real world applications, $\\bar{L}$ consists of three servers. The reason to store the data in different servers is to allow the data to be accessed through other servers if a server disconnects from the network or fails to continue working. The larger the set $\\bar{L}$ is, the more secure the data would be. However, as the storage of servers is limited, the data is usually stored on no more than three servers. So, $\\bar{L} \\in \\{ (m_1, m_2, m_3) \\in \\mathcal{M}^3, m_1 < m_2 < m_3 \\}$, and the set of all task types is denoted by $\\mathcal{L}$. Therefore, different tasks can be labeled by the location of their associated data chunk. For example, all tasks with their stored data in servers $1, 2,$ and $7$ are denoted by type $\\bar{L} = \\{ 1, 2, 7 \\}$ task. All servers in the set $\\bar{L}$ are named local servers to the task type $\\bar{L}$ since they keep the data needed for the task to be processed. The set of servers $\\{ m \\notin \\bar{L}: \\exists n \\in \\bar{L} \\ \\text{s.t.} \\ K(m) = K(n)  \\}$ are rack-local servers, and all other servers are named remote servers to type $\\bar{L}$ task. If server $m$ is local, rack-local, or remote to task of type $\\bar{L}$, it is denoted by $m \\in \\bar{L}$, $m \\in \\bar{L}_k$, or $m \\in \\bar{L}_r$, respectively. The system model is assumed to be discrete-time. If task $\\bar{L}$ is scheduled to server $m \\in \\bar{L}$ ($\\bar{L}_k$, or $\\bar{L}_r$), the probability that the task receives service in a time slot and departs the system at the end of the time slot is $\\alpha$ ($\\beta$, or $\\gamma$), where $\\alpha > \\beta > \\gamma$. In other words, the local, rack-local, and remote services follow $Geo(\\alpha)$, $Geo(\\beta)$, and $Geo(\\gamma)$, respectively. As a result, on average it takes less time for a task to receive service from a local server ($\\frac{1}{\\alpha}$), other than a rack local-local server ($\\frac{1}{\\beta}$), or a remote server ($\\frac{1}{\\gamma}$). On the other hand, the arrival of type $\\bar{L}$ task at the beginning of time slot $t$ is denoted by $A_{\\bar{L}}(t)$ which is bounded with average arrival $\\lambda_{\\bar{L}}$. The arrival of type $\\bar{L}$ tasks is assumed to be i.i.d. \\newline\nWith the service time distribution illustrated above, when a new task arrives, there might not be no local, rack-local, or remote available servers to serve the task immediately. Therefore, multiple queues exist in the data centers where the tasks are kept, waiting to receive service. Based on the structure of the data center and the scheduling algorithm, the number of queues can be less than, equal, or larger than the number of servers. For example, FIFO algorithm just needs one single queue for any number of servers to be implemented. In the rest of the paper, we will point out the number of queues needed for different algorithms. \\newline\nA question that might be raised here is \"At which queue should a new arriving task wait at so to finally receive service from a server?\". This part is handled by a \\textit{routing} algorithm which takes care of routing new tasks to appropriate queue. On the other hand, when a server is done with processing of a task and becomes idle, it needs to decide which task to pick among all tasks queued at all servers. The act of assigning tasks to idle servers is called \\textit{scheduling} with a little bit abuse of terminology. Therefore, an algorithm is fully defined by both its routing and scheduling policies. \\newline\nAs a new terminology, three levels of data locality refers to the case of having all kinds of local, rack-local, and remote services in the system. The number of data locality levels depends directly on the structure of the system. For example, if tasks receive their services just locally or remotely (no rack structure exists), then just two levels of data locality exists.\n\n\\subsection{{Capacity Region Realization}}\nLet $\\lambda_{\\bar{L}, m}$ denote the arrival rate of type $\\bar{L}$ tasks that receive service from the $m$-th server. In fact, $\\lambda_{\\bar{L}, m}$ is the decomposition of $\\lambda_{\\bar{L}}$. Assuming that a server can afford at most load 1 for all local, rack-local, and remote tasks, the capacity region can be characterized as follows \\cite{BalancedPandas, yekkehkhany2017near}:\n\n\\begin{equation}\n\\begin{aligned}\n\\Lambda = & \\{  \\boldsymbol{\\lambda} = (\\lambda_{\\bar{L}} : \\bar{L} \\in \\mathcal{L}) \\ | \\ \\exists \\lambda_{\\bar{L}, m} \\geq 0, \\forall \\bar{L} \\in \\mathcal{L}, \\forall m \\in \\mathcal{M}, s.t. \\\\\n& \\lambda_{\\bar{L}} = \\sum_{m = 1}^{M} \\lambda_{\\bar{L}, m}, \\ \\forall \\bar{L} \\in \\mathcal{L}, \\\\\n& \\sum_{\\bar{L}: m \\in \\bar{L}} \\frac{\\lambda_{\\bar{L}, m}}{\\alpha} + \\sum_{\\bar{L}: m \\in \\bar{L}_k} \\frac{\\lambda_{\\bar{L}, m}}{\\beta} + \\sum_{\\bar{L}: m \\in \\bar{L}_r} \\frac{\\lambda_{\\bar{L}, m}}{\\gamma} < 1, \\forall m \\}\n\\end{aligned}\n\\end{equation}\n\nA more thorough look into the definition of the capacity region $\\Lambda$, it is easy to figure out that for finding the capacity region of the system described in Section \\ref{systemmodel}, a linear programming optimization must be solved.\n\n\\section{Affinity Scheduling}\n\\label{affinity}\nThe near-data scheduling problem is a special case of affinity scheduling \\cite{afs1}, \\cite{mandelbaumstolyar}, \\cite{intro6}, \\cite{afs4}, and \\cite{afs5}. In this section, two algorithms, Fluid Model Planning and Generalized c$\\mu$-Rule, are illustrated which are somehow the pioneer works on the scheduling problems. However, they are not practical to be used in data centers as it will be discussed in the following two subsections.\n\\subsection{{FLUID MODEL PLANNING}}\nFor routing tasks and scheduling servers, fluid model planning algorithm is proposed by Harrison and Lopez \\cite{intro5} and \\cite{intro6} which is both throughput and heavy-traffic optimal not only for three levels of data locality, but also for the affinity scheduling problem. To implement this algorithm, distinct queues are needed for different types of tasks, $\\bar{L}$. Each new incoming task is routed to the queue of its own type. In the model described, there are at most in the order of $M^3$ types of tasks (the data associated to each task type is located on 3 servers, so at most there are ${M}\\choose{3}$ $= O(M^3)$ different task types). For finding the scheduling policy, the arrival rate of each task type is required to be known in advance. By solving a linear programming problem, the algorithm distinguishes basic activities. Based on the basic activities derived from the linear programming part, tasks are assigned to idle servers. There are two main objections for this algorithm: First, there should be in the order of $M^3$ number of queues for each task type. In practice, it is not practical to have queues in the cubic order of the number of servers. It excessively complicates the system and its underlying network. Second, the algorithm assumes the arrival rate of different types of tasks to be known. However, firstly the load is not known in real applications, and secondly it changes over time. Therefore, unless the algorithm is throughput and heavy-traffic optimal, it cannot be used in real applications.\n\n\\subsection{Generalized c$\\mu$-Rule}\nStolyar \\cite{stolyar} and Mandelbaum and Stolyar \\cite{mandelbaumstolyar} proposed Generalized c$\\mu$-Rule. On contrast to fluid model planning which uses the knowledge of the arrival rate of each task type, generalized c$\\mu$-rule uses MaxWeight notion which makes the algorithm needless of knowing the arrival rates. But similar to fluid model planning, the algorithm needs one queue per task type. For routing part, each incoming task joins the queue of its own type. Assume that the cost rate incurred by the type $\\bar{L}$ tasks queued is $C_{\\bar{L}}(Q_{\\bar{L}})$, where $Q_{\\bar{L}}$ is the queue length of type $\\bar{L}$ tasks, and the cost may generally depend on the task type. The cost function should have fairly normal features among which we can mention the followings: $C_{\\bar{L}}(.)$ is convex and continuous with $C_{\\bar{L}}(0) = 0$, $C^{'}_{\\bar{L}}(.)$ to be strictly increasing and continuous with $C^{'}_{\\bar{L}}(0) = 0$. Having the cost functions for different kinds of tasks, the server $m \\in \\mathcal{M}$ is scheduled to a task type $\\bar{L}$ in the set below when it becomes idle:\n\n\\begin{equation}\n\\begin{aligned}\n\\bar{L} \\in \\underset{\\bar{L}}{ArgMax} \\ \\bigg \\{ C^{'}_{\\bar{L}} (Q_{\\bar{L}}) \\mu_{\\bar{L}, m} \\bigg \\}\n\\end{aligned}\n\\end{equation}\n\nWhere $\\mu_{\\bar{L}, m}$ is $\\alpha$, $\\beta$, or $\\gamma$ if the task type $\\bar{L}$ is local, rack-local, or remote to the idle server $m$ respectively. For instance, if the holding cost for type $\\bar{L}$ task is $\\gamma_{\\bar{L}}Q_{\\bar{L}}^{\\beta + 1}$ with $\\beta > 0$ where satisfies all the conditions for a valid cost function, the generalized c$\\mu$-rule is proved by Stolyar to asymptotically minimize the holding cost below \\cite{stolyar}:\n\n\\begin{equation}\n\\begin{aligned}\n\\sum_{\\bar{L}} \\gamma_{\\bar{L}}^{ } Q_{\\bar{L}}^{\\beta + 1}\n\\end{aligned}\n\\end{equation}\n\nAs the constant $\\beta$ should be strictly positive in order that the cost function satisfies the conditions, the algorithm is not heavy-traffic optimal in the sense defined in Section \\ref{introduction}. Besides, using generalized c$\\mu$-rule, we still need $O(M^3)$ number of servers (where M is the number of servers). However, it is not practical having a large number of queues as the system becomes more complicated. \\newline\nAll the algorithms given in the next sections do not employ the knowledge of arrival rate, and they assume the system to have the same number of queues as the number of servers, which is a more realistic structure.\n\n\\section{{TWO LEVELS OF DATA LOCALITY}}\n\\label{twolevel}\n The model described in section \\ref{systemmodel} is for the case of three levels of data locality, as a task can be served with three different service rates $\\alpha$, $\\beta$, and $\\gamma$. However, most previous theoretical work, except the very last one by Xie, Yekkehkhany, and Lu \\cite{BalancedPandas}, has been done on two levels of data locality which is actually the base of three or more levels of data locality. The model for two levels of data locality is somehow the same as the one described in section \\ref{systemmodel}; except, in two levels of data locality, there is no notion of rack and rack-local service. Assuming the two levels of data locality, tasks either get service from a server in the set $\\bar{L}$ locally with rate $\\alpha$, or get service from any other servers remotely with rate $\\gamma$. Therefore, the capacity region would be revised as $\\sum_{\\bar{L}: m \\in \\bar{L}} \\frac{\\lambda_{\\bar{L}, m}}{\\alpha} + \\sum_{\\bar{L}: m \\notin \\bar{L}} \\frac{\\lambda_{\\bar{L}, m}}{\\gamma} < 1, \\forall m \\in \\mathcal{M}$. For two levels of data locality Wang et al. \\cite{MaxWeight}, and Xie, and Lu \\cite{Pandas} respectively proposed JSQ-MaxWeight, and Pandas algorithms that are discussed in the next two subsections below.\n\n\\subsection{{Join-the-Shortest-Queue-MaxWeight (JSQ-MW)}}\nFor two levels of data locality, JSQ-MW has been proven to be throughput optimal, but heavy-traffic optimal just in a specific traffic scenario \\cite{MaxWeight}. Wang et al. \\cite{MaxWeight} assume one queue per server, where the length of queue $m$ at time $t$ is denoted by $Q_m(t)$. A central scheduler maintains the lengths of all queues to decide the routing for new incoming tasks, and scheduling for idle servers. As of routing policy, when a new task of type $\\bar{L}$ arrives to the system, the central scheduler routes the task to the shortest queue of the servers in the set $\\bar{L}$ (all ties are broken randomly all over this paper). In other words, the new task is routed to the shortest local queue. For scheduling policy, as server $m$ becomes idle, the central scheduler assigns a task queued at a queue in the set below to server $m$:\n\n\\begin{equation}\n\\underset{n \\in \\mathcal{M}}{ArgMax} \\ \\{ \\alpha Q_n(t) I_{\\{ n=m \\}}, \\beta Q_n(t) I_{\\{ n \\neq m \\}} \\}\n\\end{equation}\n\nTherefore, the idle server gives its next available service to the queue with the maximum weight as defined above. As it was stated before, JSQ-MW is not heavy-traffic optimal in all loads. The specific traffic scenario which the JSQ-MW is heavy-traffic optimal is pointed out in section \\ref{evaluation}. For more details refer to \\cite{BalancedPandas, yekkehkhany2017near}, and \\cite{MaxWeight}.\n\nNext, Pandas algorithm proposed by Xie, and Lu \\cite{Pandas} is presented, which is both throughput and heavy-traffic optimal.\n\n\\subsection{{Priority Algorithm for Near-Data-Scheduling (Pandas)}}\nPandas algorithm is both throughput optimal and heavy-traffic optimal in all loads \\cite{Pandas}. Again, assuming one queue per server, Pandas algorithm routes the new incoming task to the shortest local queue (which is the same as JSQ-MW routing). For scheduling, an idle server always give its next service to a local task queued at its own queue; unless, its queue length is zero. If the idle server's queue does not have any tasks queued, the central scheduler assigns a task from the longest queue in the system to the idle server (which the task is remote to the idle server). This assignment of the remote task from the longest queue in the system occurs when $Q_{max}(t) \\geq \\frac{\\alpha}{\\gamma}$, to make sure that the remote task experiences less service time in the remote server other than waiting and receiving its service from the local server.\n\n\nAs the conclusion of the previous work for two levels of data locality, Pandas algorithm proposed by Xie, and Lu \\cite{Pandas} is the most promising algorithm that both stabilizes the system in the whole capacity region, and minimizes mean delay for all tasks in heavy-traffic regime. However, in real applications there are usually more than two levels of data locality. The reason is that a server may have the data stored on the memory or local disk, or the server may not have the data saved locally, so it has to fetch the data from another server in the same rack, or even from another server in other racks, which results in appearance of multi levels of data locality. Therefore, it is more of interest to come up with a throughput and heavy-traffic optimal algorithm for a system with more than two levels of data locality. The model illustrated in the system model section has three levels of locality, as a task can get its service locally, rack-locally, or remotely. For the purpose of designing a throughput and heavy-traffic optimal algorithm, assuming three levels of data locality is more challenging than two levels, as a trade-off between throughput optimality and delay optimality emerges. The Pandas algorithm proposed by Xie, and Lu \\cite{Pandas} which is both throughput and heavy-traffic optimal for two levels of data locality is not even throughput optimal for three levels of data locality. Xie, Yekkehkhany, and Lu \\cite{BalancedPandas} proposed two algorithms for three levels of data locality which are discussed in the next section.\n\n\\section{{THREE LEVELS OF DATA LOCALITY}}\n\\label{threelevel}\nFor three levels of data locality of which the system model is described in section \\ref{systemmodel}, Xie, Yekkehkhany, and Lu \\cite{BalancedPandas} extended the JSQ-MaxWeight algorithm and proved it to be throughput optimal. However, the extension of JSQ-MaxWeight is still heavy-traffic optimal only in a specific traffic scenario, not in all loads (note again that JSQ-MW is not also heavy-traffic optimal in all loads for two level of data locality, except a specific load). Xie et al. \\cite{BalancedPandas} also proposed a new algorithm called the weighted-workload routing and priority scheduling algorithm which is throughput optimal for any $\\alpha > \\beta > \\gamma$, and is heavy-traffic optimal for the case that $\\beta^2 > \\alpha \\gamma$, which usually holds in real data centers. What $\\beta^2 > \\alpha \\gamma$ implies is that the rack-local service is much faster than remote service. In the next two subsections, JSQ-MaxWeight and the weighted-workload routing and priority scheduling algorithm are discussed in more details.\n\n\\subsection{{Extension of JSQ-MaxWeight}}\nAssuming one queue per server, the JSQ-MW is as follows: \\\\\n\\textbf{Routing:} An arriving task is routed to the shortest queue of the servers in the set $\\bar{L}$ (shortest local queue). \\\\\n\\textbf{Scheduling:} An idle server is scheduled to a queue in the set\n\\begin{equation}\n\\begin{aligned}\n\\underset{n \\in \\mathcal{M}}{ArgMax} \\ \\{ & \\alpha Q_n(t) I_{\\{ n = m \\}}, \\beta Q_n(t) I_{\\{ K(n) = K(m) \\}},\\\\\n&\\gamma Q_n(t) I_{\\{ K(n) \\neq K(m) \\}} \\}\n\\end{aligned}\n\\end{equation}\n\n\n\\begin{theorem}\nJSQ-MaxWeight stabilizes the system under any arrival rate vector within $\\Lambda$. Therefore, JSQ-MaxWeight is throughput optimal.\n\\end{theorem}\n\nProof outline. If $\\boldsymbol{\\lambda} \\in \\Lambda$, using $V(t) = \\sum_{n = 1}^{M} Q_n^2(t)$ as the Lyapunov function, the $T$ time slot drift of $V(t)$ is bounded in a finite subset of state space of the system, and is negative outside of this subset, which results in stability of the system by Foster-Lyapunov theorem \\cite{BalancedPandas, yekkehkhany2017near} (you can refer to \\cite{MaxWeight, Pandas, ghassami2017covert} for other uses of Foster-Lyapunov theorem for stability proof of a system).\n\n\\begin{theorem}\nThe extended JSQ-MaxWeight is heavy-traffic optimal in a special scenario of the workload, but not in all traffic scenarios.\n\\end{theorem}\n\n\\subsection{{Weighted Workload Routing and Priority Scheduling}}\nAlthough it is sufficient to have one queue per server to implement the weighted-workload routing and priority scheduling algorithm in a data center with three levels of data locality, it is easier to describe this algorithm assuming the existence of three queues per server. Therefore, assume each server to have three queues, which local, rack-local, and remote tasks to the server are queued in the three queues separately. The central scheduler keeps the vector of queue lengths $\\boldsymbol{Q}(t) = (\\boldsymbol{Q}_1(t), \\boldsymbol{Q}_2(t), \\dots, \\boldsymbol{Q}_M(t))$ where $\\boldsymbol{Q}_m(t) = (Q_m^l(t), Q_m^k(t), Q_m^r(t))$. That is, the first queue of $m$-th server keeps the tasks that are routed to server $m$ and are local to it, the second queue of the $m$-th server keeps the tasks that are routed to this server, but are rack-local to it, and finally remote tasks to the $m$-th server that are routed to this server are queued in the third queue.\n\nDefining the workload on a server, it is ready to give the routing and scheduling policies. The workload on the $m$-th server is defined as the average time needed for the server to give service to all local, rack-local, and remote tasks queued in front of it. The workload on the $m$-th server is defined below:\n\n\\begin{equation}\nW_m(t) = \\frac{Q_m^l(t)}{\\alpha} + \\frac{Q_m^k(t)}{\\beta} + \\frac{Q_m^r(t)}{\\gamma}\n\\end{equation}\n\n\n\\textbf{Weighted-Workload Routing:}\nAs a new task arrives, it joins the server with the least weighted workload. More precisely, the new task joins one of the servers in the following set, where the ties are broken randomly.\n\\begin{equation}\n\\underset{m \\in \\mathcal{M}}{ArgMin} \\ \\bigg \\{ \\frac{W_m(t)}{\\alpha} I_{\\{ m\\in \\bar{L} \\}},\\frac{W_m(t)}{\\beta} I_{\\{ m\\in \\bar{L}_k \\}}, \\frac{W_m(t)}{\\gamma} I_{\\{ m\\in \\bar{L}_r \\}}  \\bigg \\}\n\\end{equation}\nIf the task is local (rack-local, or remote) to the server with the least weighted workload, it joins the first (second, or third) queue which is $Q_m^l$ ($Q_m^k$, or $Q_m^r$).\n\n\\textbf{Priority Scheduling:}\nWhen a server, say $m$, becomes idle, it gives its next service to local tasks queued in front of it at $Q_m^l$. In case that there is no local task available for idle server $m$, that is $Q_m^l = 0$, next service is assigned to rack-local tasks queued at $Q_m^k$. Finally if both local, and rack-local queues are empty, the next service goes to $Q_m^r$. In summary, the idle server gives the most priority to local, then rack-local, and finally remote tasks. If all three sub-queues of server $m$ are empty, the server remains idle; until, a new task joins any of the three sub-queues.\n\n\\begin{theorem}\nThe weighted-workload routing and priority scheduling algorithm is throughput optimal, as it stabilizes any arrival rate vector in the capacity region.\n\\end{theorem}\n\nProof outline. The $T$ time slot drift of the following Lyapunov function is bounded in a finite subset of state space, and is negative out of this subset, as long as the arrival rate vector is in the capacity region ($\\boldsymbol{\\lambda} \\in \\Lambda$), which results in the stability of the system \\cite{BalancedPandas, yekkehkhany2017near}.\n\\begin{equation}\nV(t) = ||\\boldsymbol{W}||^2 = \\sum_{m \\in \\mathcal{M}} \\bigg ( \\frac{Q_m^l(t)}{\\alpha} + \\frac{Q_m^k(t)}{\\beta} + \\frac{Q_m^r(t)}{\\gamma} \\bigg )^2\n\\end{equation}\n\n\\begin{theorem}\n\\label{htobp}\nThe weighted-workload routing and priority scheduling algorithm is heavy-traffic optimal for $\\beta^2 > \\alpha \\gamma$ \\cite{BalancedPandas}.\n\\end{theorem}\n\n\n\n\n\\section{{CONCLUSION}}\n\\label{conclusion}\nIn this paper, we first discussed the history of task routing and affinity scheduling in data centers. Two algorithms are then proposed for a system with two levels of data locality: JSQ-MaxWeight, and Pandas algorithms for Near-Data Scheduling (Pandas)- both of which are throughput optimal. However, it was shown that Pandas is the only algorithm being heavy-traffic optimal in all loads. Taking further steps to three levels of data locality, we mentioned that the Pandas algorithm known to be heavy-traffic optimal for two levels of data locality is not even throughput optimal for three levels of data locality. Then, an algorithm with weighted workload routing and priority scheduling as well as an extension of JSQ-MaxWeight are discussed for three levels of data locality. Among these two algorithms only the weighted-workload routing and priority scheduling algorithm is heavy-traffic optimal in all loads. \n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCryptosystems based on lattices are one of the leading alternatives to RSA and ECC that are conjectured to be resistant to quantum attacks. Considered to be the genesis of the study of lattice cryptosystems, in 1996, Ajtai constructed a one-way function, and proved the average-case security related to the worst-case complexity of lattice problems \\cite{ajtai}. Later, in 2005, Regev proposed the computational problem known as ``Learning With Errors'' (LWE) and showed that LWE is as hard to solve as several worst-case lattice problems \\cite{LWE}. Whilst these problems are believed to be difficult to crack even given access to a quantum computer, their main drawback is their impracticality to implement in cryptosystems due to the large key sizes required to define them.\n\\\\ \\indent For this reason, lattices with algebraic structure are often favoured to define cryptosystems over conventional lattices. In particular, cryptosystems often employ the use of cyclotomic polynomials as their cyclic nature allows for much less cumbersome computations. Such lattices come in two main varieties: ideal lattices, whose structures are formed entirely by embedding an ideal of a number field into real or complex space, and module lattices, which are free modules defined over an algebraic ring and can be thought of as a compromise between classical and ideal lattices. Perhaps the most well-known cryptosystem based on algebraic structure is the NTRU cryptosystem. Developed in 1996 by Hoffstein, Pipher and Silverman \\cite{NTRU}, the NTRU cryptosystem uses elements of the convolution ring $\\mathbb{Z}[x]\/(x^p-1)$ and offers efficient encryption and decryption of messages, making it one of the most popular lattice-based cryptosystems even to this day. In \\cite{securentru}, noting that the ring $\\mathbb{Z}[x]\/(x^p-1)$ could be deemed insecure due to the fact that $x^p-1$ is not irreducible, Stehl\\'e and Steinfeld updated NTRU to incorporate a cyclotomic ring in place of the aforementioned ring. The computational problem involved in breaking NTRU can be thought of as a rank 2 module problem over a cyclotomic ring. More recently, an algebraic variant of LWE called Ring-LWE (RLWE) was developed by Lyubashevsky, Peikert and Regev in 2010 \\cite{RLWE}. It has been shown that the security of this scheme relies heavily on the hardness of ideal lattice reduction \\cite{RLWE-hardness}. Moreover, the work by Ajtai was also generalised to the ring case by Micciancio in 2004 \\cite{knapsack}. Using an arbitrary ring in place of a classical lattice, he managed to show that obtaining a solution to the ring-based alternative to the knapsack problem on the average was at least as hard as the worst-case instance of various approximation problems over cyclic lattices, even for rings with relatively small degree over $\\mathbb{Z}$. Whilst there are a myriad of other schemes that make use of algebras to define a cryptosystem (see for example \\cite{newhope}, \\cite{kyber}, \\cite{falcon}), concerns have been raised regarding the security of such schemes. Whilst the algebraic structure might allow for easier computations, the additional structure given by an algebra could be exploited to allow for easier reduction of lattices based on algebras.\n\\\\ \\indent As we have already mentioned, cyclotomic polynomials exhibit many properties that are desirable in cryptography. In particular, cryptographers largely favour power-of-two cyclotomic rings, that is, cyclotomic rings with conductor $N=2^n$ for some integer $n$. This is largely a consequence of a few properties exhibited by power-of-two cyclotomic rings, for example, $N\/2$ is also a power of two and arithmetic in the ring can be performed with ease using the $N$-dimensional FFT. However, restricting cryptosystems to only using power-of-two cyclotomic rings has its drawbacks. The most obvious of these drawbacks is the increase in dimension of the ideal lattice when moving from one power-of-two cyclotomic ring to the next, which doubles with each successive ring, and the cryptosystem may require a lattice of intermediate security: for example, ideal lattices of the cyclotomic ring of conductor $1024$ have dimension $512$, but the next power-of-two cyclotomic ring is of conductor $2048$, and so ideal lattices defined in this ring have dimension $1024$, which is a significant jump. For this reason amongst others, cryptographers have begun to move away from power-of-two cyclotomic rings to cyclotomic rings of more general conductor. However, this migration from power-of-two cyclotomics is relatively novel, and as such literature regarding the reduction of ideal lattices based on cyclotomic rings of general conductor is still heavily lacking.\n\\subsection{Previous Works}\nThere have been a variety of studies into the shortest vector problem (SVP) in lattices generated by ideals. In 2016, Cramer, Ducas, Peikert and Regev published a paper detailing an attack on ideals generated by principal ideals of prime-power cyclotomic rings \\cite{shortgen}. They presented a technique involving the use of the log-unit lattice, and showed that there is a polynomial time classical reduction from the problem of finding a short generator of a principal ideal (SPIP) to the problem of finding a generator of a principal ideal (PIP).  Moreover, in \\cite{biasset}, it was shown how to solve PIP in polynomial time with a quantum computer by Biasset and Song. Thus, combining these results, there is a polynomial time quantum algorithm for SPIP, and we note that SPIP corresponds to finding approximate short vectors in principal ideals. Later, Cramer, Ducas, and Wesolowski gave a quantum (heuristic) reduction of the general approx-SVP algorithm to SPIP \\cite{stickelb}. Therefore a quantum polynomial algorithm for approx-SVP follows from the above three papers.\\\\\n\\indent Simultaneously, largely inspired by Bernstein's work on subfield attacks against ideal lattices \\cite{subfieldlogarithm}, Albrecht, Bai and Ducas proposed a different method by which to attack the NTRU cryptosystem with overstretched parameters - that is, the NTRU encryption scheme with much larger modulus \\cite{subfield}. Their method entailed an attack on the NTRU cryptosystem by attacking a sublattice, defined by a public key attained by ``norming down'' the public key of the original lattice to a subfield, and then ``lifting'' a solution on the sublattice to a solution for the original cryptosystem which, provided the solution is sufficiently good in the sublattice, may yield a short lattice vector in the full lattice. Indeed, there are many examples of previous works which detail lattice attacks against ideal lattices. For a detailed list regarding previous research into the reduction of ideal lattices, we refer the reader to the ``previous works'' section of \\cite{idealrandomprime}.\n\\\\ \\indent Recently, Pan, Xu, Wadleigh and Cheng pioneered a remarkable technique to approach the problem of SVP in prime and general ideal lattices, obtained from power-of-two cyclotomics \\cite{idealrandomprime}. Their method involved manipulating the decomposition group of prime ideals in order to significantly reduce the dimension of the lattice required to solve SVP for. Their primary contributions were in two parts: the first part of the paper took a number field $L$ that is Galois over $\\mathbb{Q}$ and showed that, given a prime ideal of its ring of integers $\\mathcal{O}_L$, if Hermite-SVP can be solved for a certain factor in a sublattice generated by a subideal, this yields a solution for Hermite-SVP in the original ideal lattice with a larger factor, where the factor's increase depends only on the square root of the degree of $L$ over $\\mathbb{Q}$ divided by the size of the decomposition group. The second part of their paper is dedicated to ideals over the ring of integers of cyclotomic fields of conductor $N=2^{n+1}$. Under the so-called coefficient embedding, they showed that using a subgroup of the decomposition group of a prime ideal, the shortest vector in the ideal is equivalent to the shortest vector in a subideal constructed in the paper, and so solving SVP over such ideals is easy given an oracle to solve SVP in lattices of dimension equivalent to the dimension of the ideal lattice generated by the subideal. Moreover, they showed that if such a prime ideal lies above a rational prime $p$ of the form $p \\equiv \\pm 3 \\mod 8$ then the shortest vector is of length $p$, and is very easy to determine. In the final section, they showed that their method also worked for general ideals by considering the prime decomposition of an ideal.\n\\subsection{Our Results}\nIn this paper, we generalise the results of Pan et. al., both in their work on Hermite-SVP for prime ideal lattices and solving SVP exactly for prime ideals in cyclotomic ideals. The first half of the paper is dedicated to the Hermite-SVP in ideal lattices. Whilst Pan et. al. only covered the case for lattices based on prime ideals lying above unramified primes, we extend their result to the case of general ideals whose prime ideal factors all lie over unramified primes, showing that by solving Hermite-SVP on a subideal with some factor $\\gamma$, the solution may be lifted to yield a solution for Hermite-SVP in the original lattice with factor $\\gamma^{\\prime}$, where $\\gamma^{\\prime}\/\\gamma$ depends only on the factor given by Pan et. al. multiplied by a value determined by certain properties of the ideal and its decomposition group. We take this notion even more generally, and consider a module over the ring of integers of a Galois field and provide a method where a solution to Hermite-SVP in a submodule generating a lattice of lower dimension may be lifted to a solution in the original module lattice for Hermite-SVP with an upper bound, where the new constant is given in terms of the old factor multiplied by some factor dependent only on the ideals used to describe the module in the pseudo-basis representation.\n\\\\ \\indent The second half of the paper focuses on prime ideals of cyclotomic rings. Our work extends the results of Pan et. al. to prime ideal lattices constructed from more cyclotomic rings of more general conductors, covering the cases of a general composite conductor $N=s2^{n+1}$ and $s^{\\prime} p^{n+1}$ for some odd prime $p$, odd integer $s \\geq 3$ and integer $s^{\\prime}$, $\\gcd(s^{\\prime},p)=1$, which, combined with the work of Pan et. al., covers the case for any conductor $N$. In particular, our work shows that if the prime ideal in question lies above certain primes, then the dimension in which we have to solve SVP decreases significantly.\n\\begin{theorem}\nLet $N=s2^{n+1}$, where $n$ is a positive integer and $s \\geq 3$ is an odd integer. Let $\\mathfrak{p}$ be a prime ideal in the ring $\\mathbb{Z}[\\zeta_N]$ and suppose that $\\mathfrak{p}$ contains a rational prime $\\rho$, where $\\rho^{\\phi(s)} \\equiv 3 \\mod 4$. Then, given an oracle that can solve SVP for $\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in time $\\text{poly}(\\phi(N),\\log_2 \\rho)$ under the canonical embedding.\n\\end{theorem}\n\\begin{theorem}\nLet $N=sp^{n+1}$, where $n$ is a positive integer, $p$ is an odd prime and $s$ is a positive integer such that $\\gcd(s,p)=1$. Let $\\mathfrak{p}$ be a prime ideal in the ring $\\mathbb{Z}[\\zeta_N]$ and suppose that $\\mathfrak{p}$ contains a rational prime $\\rho$, where $\\rho^{\\phi(s)} =lp + a$ for some integers $l,a$, $\\gcd(l,p)=\\gcd(a,p)=1$. Then, given an oracle that can solve SVP for $(p-1)\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in time $\\text{poly}(\\phi(N),\\log_2 \\rho)$ under the canonical embedding.\n\\end{theorem}\n\\noindent As with the case for conductor $N=2^{n+1}$, we must ask whether the ``average case'' of prime ideal SVP over such cyclotomic rings is easy. This question in itself is ill-defined, and depends on how we define the distribution from which we choose the prime ideal. As we show in section 5, if we were to pick our ideal by uniformly choosing an ideal from the set of prime ideals whose rational prime lies below a certain bound, the probability of choosing an easily solvable ideal lattice is non-negligible. However, if we are to uniformly choose from the distribution of ideals of norm less than a certain bound, the probability of choosing an easily solvable ideal lattice is negligible.\n\\\\ \\indent In the last few sections, we also cover the case of general cyclotomic ideals and modules defined over a pseudo basis of ideals and vectors. For the case of general ideals, in a similar fashion to that in Pan et. al.'s work, we analyse SVP by studying the prime decomposition of ideals, and show that the shortest nonzero vector in a general ideal can be found by finding the shortest nonzero vector in a subideal of a smaller dimension. Moreover, the algorithm used to tackle SVP in such a lattice does not use the prime decomposition of the ideal, which is the most computationally complex step after SVP. In the module case, we do not explicitly construct an algorithm to perform SVP, however we show that using the structure theorem for finitely generated modules over a principal ideal domain it is possible to construct an isomorphism such that SVP in the original module can be found by finding the shortest nonzero vector in a module which has smaller dimension as a lattice after canonically embedding the module.\n\\\\ \\indent Whilst the work on ideal-SVP may initially appear to destabilise the security of cryptosystems based on cyclotomic ideals, we must point out that this work does not break Ring-LWE. Though Ideal-SVP underpins the security of Ring-LWE, our work does not directly impact the security of these schemes, since the worst-case to average-case security reduction is one-way. However, our results on module SVP may cause concern for the security of MLWE, and hence RLWE. Unlike ideal SVP and RLWE, module SVP and MLWE are known to be (polynomially) equivalent, as are MLWE and RLWE. In section 6, we offer an algorithm that can be used to find the shortest vector in a general module lattice over a cyclotomic ring by solving SVP in a submodule defined over a cyclotomic ring of lesser degree, which decreases the dimension of the lattice required to perform SVP over under the canonical embedding significantly in some cases.\n\\subsection{Paper organisation}\nThe paper is organised as follows. Section 2 covers the mathematical preliminaries, including a definition of lattices and their various properties, some basic algebraic number theory and ideal lattices. The preliminary section ends with some useful lemmas regarding the factorisation of polynomials over finite fields. Section 3 covers a reduction Hermite-SVP in ideal lattices and module lattices based over a Galois extension of $\\mathbb{Q}$. In section 4, we present a reduction of SVP for prime ideals of cyclotomic rings of general conductor, and show that some special cases of prime ideals are much easier to perform SVP for than others. In section 5, we show that our method for prime ideal lattices may be lifted to the case of general ideal lattices of cyclotomic rings. In section 6, we show that modules over cyclotomic rings may be subject to a similar reduction of SVP by studying the decomposition group of the module, and provide an algorithm which can be used to solve SVP in module lattices over cyclotomic rings. In section 7, we discuss the average-case hardness of SVP for ideal and module lattices of cyclotomic rings. \n\\section{Mathematical Preliminaries}\n\\subsection{Lattices and the Shortest Vector Problem}\nA lattice is a discrete additive subgroup of $\\mathbb{R}^D$. A lattice $L$ has a basis $B=\\{\\mathbf{b}_1,\\dots,\\mathbf{b}_d\\}, \\mathbf{b}_i \\in \\mathbb{R}^D$ for some integer $d \\leq D$, and every lattice point may be represented by the linear sum of basis vectors over the integers, that is,\n\\begin{align*}\n    L=L(B)=\\left\\{\\sum_{i=1}^dx_i \\mathbf{b}_i: x_i \\in \\mathbb{Z}\\right\\}.\n\\end{align*}\nWe say that $L$ is full-rank if $d=D$. The determinant of $L$, $\\det(L)$, is the square root of the volume of the fundamental parallelopiped generated by the lattice basis. If $L$ is full-rank, then $\\det(L)=|\\det(B)|$.\n\\\\\nIn cryptography, the security of a lattice-based cryptosystem in most cases boils down to the computational hardness of the shortest vector problem (SVP). The problem goes as follows: given a lattice $L$ with basis $B$, find the shortest nonzero vector in $L$ with respect to the Euclidean (or otherwise specified) norm. Most cryptosystems, however, loosen the requirement of finding the shortest nonzero vector, and require the assailant to find a nonzero vector within some range of the shortest vector. One such problem is known as the Hermite-SVP, and goes as follows.\n\\begin{definition}\nLet $L$ be a rank $N$ lattice. The $\\gamma$-Hermite-SVP is to find a nonzero lattice vector $\\mathbf{v} \\in L$ that satisfies\n\\begin{align*}\n    \\|\\mathbf{v}\\| \\leq \\gamma \\det(L)^{1\/N},\n\\end{align*}\nfor some approximation factor $\\gamma \\geq 1$.\n\\end{definition}\n\\noindent As opposed to the shortest lattice vector, the determinant of a lattice is well-defined and can be verified easily. Moreover, as discussed in \\cite{idealrandomprime}, a solution to Hermite-SVP can be lifted to a solution for a variety of different SVP-related problems.\n\\subsection{Algebraic Number Theory}\nAn algebraic number field $L$ is a finite extension of $\\mathbb{Q}$ by some algebraic integer $\\alpha$, that is, the solution to a polynomial in $\\mathbb{Z}[x]$. The degree of $L$ over $\\mathbb{Q}$ is equivalent to the degree of the minimal polynomial of $\\alpha$ in $\\mathbb{Q}(x)$. We denote by $\\mathcal{O}_L$ the ring of integers of $L$, which is the maximal order of $L$. It is well known in algebraic number theory that any algebraic number field $L$ is a $\\mathbb{Q}$-vector space over the power basis $\\{1,\\theta,\\theta^2,\\dots, \\theta^{N-1}\\}$ for some $\\theta \\in L$, and similarly the ring of integers $\\mathcal{O}_L$ may be expressed as a $\\mathbb{Z}$-module over a power basis $\\{1,\\theta^{\\prime},\\dots, {\\theta^{\\prime}}^{N-1}\\}$ for some $\\theta^{\\prime} \\in \\mathcal{O}_K$ \\cite{algnumbertheory}.\n\\\\ \\indent For a positive integer $N$, the cyclotomic polynomial $\\Phi_N(x)$ is the polynomial given by\n\\begin{align*}\n    \\Phi_{N}(x)=\\prod_{k=1: \\gcd(k,N)=1}^N\\left(x-\\exp\\left(\\frac{2\\pi i k}{N}\\right)\\right),\n\\end{align*}\nor in other words, the polynomial whose roots are all the primitive $N$th roots of unity. For ease of notation, we generally let $\\zeta_N=\\exp\\left(\\frac{2\\pi i}{N}\\right)$ denote the $N$th root of unity. The field $L=\\mathbb{Q}(\\zeta_N)$ obtained by appending $\\zeta_N$ to $\\mathbb{Q}$ is called the cyclotomic field of conductor $N$, and such a field is of degree $\\phi(N)$ over $\\mathbb{Q}$, where\n\\begin{align*}\n    \\phi(N) = N\\prod_{p \\mid N: p\\hspace{0.5mm} \\text{prime}}\\left(1-\\frac{1}{p}\\right)\n\\end{align*}\nis Euler's totient function, which measures the number of integers less than or equal to $N$ which are coprime to $N$.\n\\\\ The embeddings $\\sigma$ of a number field $L$ are the injective homomorphisms from $L$ to $\\mathbb{C}$ which fix $\\mathbb{Q}$. The number of distinct embeddings is equivalent to the degree of $L$ over $\\mathbb{Q}$, and an embedding $\\sigma$ is said to be a real embedding if $\\sigma(L) \\subset \\mathbb{R}$, and is said to be a complex embedding if $\\sigma(L) \\not\\subset \\mathbb{R}$. We define the \\emph{canonical embedding} $\\Sigma_L$ from a number field $L$ of degree $N$ to $\\mathbb{C}^N$ by\n\\begin{align*}\n    \\Sigma_L: L \\to \\mathbb{C}^N \\hspace{2mm} a \\mapsto (\\sigma_1(a),\\sigma_2(a),\\dots, \\sigma_N(a)).\n\\end{align*}\nMoreover, we respectively define the trace and norm of an element in $L$ by\n\\begin{align*}\n    \\text{Trace}_{L\/\\mathbb{Q}}(a) \\vcentcolon= \\sum_{i=1}^N\\sigma_i(a), \\hspace{2mm} \\text{Norm}_{L\/\\mathbb{Q}}(a)=\\prod_{i=1}^N\\sigma_i(a).\n\\end{align*}\nDefining by $\\overline{a}$ the complex conjugate of an element $a \\in L$, note that $\\beta(x,y) \\vcentcolon= \\text{Trace}_{L\/\\mathbb{Q}}(x\\overline{y})$ for all $x,y \\in L$ defines a positive-definite bilinear form on the $\\mathbb{Q}$-vector space generated by $L$. \n\\\\ \\indent Another way of embedding a number field $L$ into $\\mathbb{C}^N$ is using the so-called \\emph{coefficient embedding}. By expressing $L$ by a $\\mathbb{Q}$-vector space over a power basis $\\{1,\\alpha,\\dots, \\alpha^{N-1}\\}$, we may take any element $a=\\sum_{i=0}^{N-1}a_i\\alpha^i$ where $a_i \\in \\mathbb{Q}$ and define the embedding map\n\\begin{align*}\n    a \\mapsto (a_0,a_1,\\dots, a_{N-1}).\n\\end{align*}\nIt is important to note that the coefficient embedding depends on the choice of basis for $L$. Also, if $\\mathcal{O}_K=\\mathbb{Z}[\\alpha]$ then $\\mathcal{O}_K$ maps to $\\mathbb{Z}^N$ under the coefficient embedding.\n\\subsection{Ideal Lattices}\nAn ideal $\\mathcal{I}$ is a subring of the ring of integers $\\mathcal{O}_L$. We say that an ideal $\\mathfrak{p}$ is prime if, for any $ab \\in \\mathfrak{p}$ for $a,b \\in \\mathcal{O}_L$, then either $a$ or $b$ is an element of $\\mathfrak{p}$. Under the canonical embedding, an ideal forms a lattice in $\\mathbb{R}^N$, and we call lattices constructed in this way \\emph{ideal lattices}. The volume of an ideal lattice $\\mathcal{I}$ in $\\mathbb{R}^N$ is $\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{I})disc(L\/\\mathbb{Q})$, where $\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{I})$ is the norm of the ideal $\\mathcal{I}$ and is equivalent to the cardinality of $\\mathcal{O}_L\/\\mathcal{I}$ (roughly speaking, the ``density'' of the ideal in $\\mathcal{O}_L$), and $disc(L\/\\mathbb{Q})$ is the discriminant of $L$ over $\\mathbb{Q}$, which is equivalent to the volume of the lattice generated after embedding $\\mathcal{O}_L$ via the canonical embedding.\n\\\\ \\indent In lattice-based cryptography, the cyclotomic number field $L=\\mathbb{Q}(\\zeta_N)$ is frequently used to define an ideal lattice. The ring of integers of $L$ is $\\mathcal{O}_L=\\mathbb{Z}[\\zeta_N]$ for any conductor $N$ \\cite{washington}. Suppose that the cyclotomic polynomial $\\Phi_N(x)$ factors in the finite field as\n\\begin{align*}\n    \\Phi_N(x)=\\prod_{i=1}^g f_i(x)^e \\mod p\n\\end{align*}\nfor some rational prime $p$, where $f_i(x)$ are irreducible mod $p$. Then the ideal $p\\mathcal{O}_L$ factors as\n\\begin{align*}\n    p\\mathcal{O}_L=(\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_g)^e,\n\\end{align*}\nwhere each $\\mathfrak{p}_i=\\langle p,f(\\zeta_N) \\rangle$ are prime ideals. We say the ideal $\\mathfrak{p}_i$ \\emph{lies over} $p$. If $e>1$, then $p$ is said to be \\emph{ramified} in $\\mathcal{O}_L$, and otherwise ($e=1$) $p$ is \\emph{unramified} in $\\mathcal{O}_L$. As such, we are motivated to study the factorisation of cyclotomic polynomials over finite fields in order to better study the structure of prime ideals. However, before we delve into more technical details regarding the factorisation of polynomials over finite fields, we introduce the following definition, which will be a recurring theme throughout the paper, and will be a powerful tool to help tackle SVP in ideal lattices.\n\\begin{definition}\nLet $L\/\\mathbb{Q}$ be a finite Galois extension of degree $N$, and let $G$ be the Galois group of $L$ over $\\mathbb{Q}$. The \\emph{decomposition group} $D$ of a prime ideal $\\mathfrak{p}$ is a subgroup of $G$ satisfying\n\\begin{align*}\n    D=\\{\\sigma \\in G: \\sigma(\\mathfrak{p})=\\mathfrak{p}\\},\n\\end{align*}\nthat is, the embeddings of $L$ that fix $\\mathfrak{p}$. Then the decomposition field $K$ of $\\mathfrak{p}$ is defined by\n\\begin{align*}\n    K=\\{x \\in L: \\forall \\sigma \\in D, \\sigma(x)=x\\},\n\\end{align*}\nthat is, the subfield of $L$ that is fixed by the decomposition group.\n\\end{definition}\n\\subsection{Factorisation of Cyclotomic Polynomials over Finite Fields}\nThe following lemmas will be used throughout section 4 onwards. Lemmas \\ref{factornumber}-\\ref{order} are standard in the study of finite fields, and we point the reader to \\cite{finitefields} for more details, and also for any terminology regarding finite fields. Lemma \\ref{pqdivide} is stated and proved in \\cite{factorpn}.\n\\begin{lemma}\\label{factornumber}\nLet $q$ be a power of a prime and $N$ be a positive integer such that $\\gcd(q,N)=1$. Then the $N$th cyclotomic polynomial $\\Phi_{N}(x)$ can be factorised into $\\phi(N)\/m$ distinct monic irreducible polynomials of the same degree $m$ over $\\mathbb{F}_q$, where $m$ is the least positive integer such that $q^m \\equiv 1 \\mod N$.\n\\end{lemma}\n\\begin{lemma}\\label{tmonic}\nLet $f_1(x),f_2(x),\\dots,f_N(x)$ be distinct monic irreducible polynomials over $\\mathbb{F}_q$ of degree $m$ and order $e$, and let $t \\geq 2$ be an integer whose prime factors divide $e$ but not $\\frac{q^m-1}{e}$. Assume that $q^m \\equiv 1 \\mod 4$ if $t \\equiv 0 \\mod 4$. Then $f_1(x^t),f_2(x^t),\\dots, f_N(x^t)$ are all distinct monic irreducible polynomials of degree $mt$ and order $et$.\n\\end{lemma}\n\\begin{lemma}\\label{order}\nLet $f(x)$ be an irreducible polynomial over $\\mathbb{F}_q$ of degree $m$ and with $f(0) \\neq 0$. Then the order of $f(x)$ is equal to the order of any root of $f(x)$ in the multiplicative group $\\mathbb{F}_{q^m}^*$.\n\\end{lemma}\n\\begin{lemma}\\label{pqdivide}\nLet $p$ be an odd prime, and $q$ be a prime power such that $q \\equiv 1 \\mod p$. If $m,n$ are positive integers satisfying $p^n \\mid q^{p^{n-m}}-1$ and $p \\nmid \\frac{q^{p^{n-m}}-1}{p^n}$, then $p^{n+1} \\mid q^{n+1-m}-1$ and $p \\nmid \\frac{q^{p^{n+1-m}}-1}{p^{n+1}}$.\n\\end{lemma}\n\\section{Solving Hermite-SVP  for general ideal lattices in a\nGalois extension}\nIn this section, we generalise the results of Pan et. al., specifically their contributions on Hermite-SVP in prime ideal lattices. We consider first general ideal lattices, and then modules with a pseudo-basis of ideals and vectors with entries in the overlying number field.\n\\begin{defn}\nLet $L\/\\mathbb{Q}$ be a finite Galois extension and $I\\subset\\mathcal{O}_L$ an ideal, expressible as $\\mathcal{I} = \\mathfrak{p}_1...\\mathfrak{p}_g$, where each $\\mathfrak{p}_i$ lies above unramified rational prime $p_i$. Let $D_\\mathcal{I} = \\{\\sigma\\in Gal(L\/\\mathbb{Q}):\\sigma(\\mathcal{I})=\\mathcal{I}\\}$, and $K_\\mathcal{I} = L^{D_\\mathcal{I}} = \\{x\\in L\\text{ : }\\sigma(x)=x, \\text{ for all }\\sigma\\in D_\\mathcal{I}\\}$. These are called the \\textit{decomposition group} and \\textit{decomposition field} of $\\mathcal{I}$, respectively.\n\\end{defn}\n\\begin{theorem}\nLet ${L} \/ \\mathbb{Q}$ be a finite Galois extension of degree $N$ and $\\mathcal{I} = \\mathfrak{p}_1...\\mathfrak{p}_g$ an ideal of $\\mathcal{O}_{{L}}$, where each $\\mathfrak{p}_i$ lies over an unramified rational prime $p_i$ such that $p_i$ has $g_i$ distinct prime ideal factors in $O_{{L}}$ and has inertial degree $f_{p_i}^L$ in $\\mathcal{O}_L$. If ${K}_\\mathcal{I}$ is the decomposition field of $\\mathcal{I}$, and each $p_i$ has inertial degree $f_{p_i}^{K_\\mathcal{I}}$ in $\\mathcal{O}_{K_{\\mathcal{I}}}$, then a solution to Hermite-$SVP$ with factor $\\gamma$ in the sublattice $\\mathfrak{c}=\\mathcal{I} \\cap \\mathcal{O}_{{K}_\\mathcal{I}}$ under the canonical embedding of ${K}_\\mathcal{I}$ will also be a solution to Hermite-SVP in $\\mathcal{I}$ with factor $\\gamma\\frac{\\sqrt{N\/r}\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{I})^{1\/r-1\/N}}{\\text{Norm}_{{K_\\mathcal{I}}\/\\mathbb{Q}}({disc}({L} \/ {K}_\\mathcal{I}))^{1\/2N}(p_1^{(f^L_{p_1}-f^{K_\\mathcal{I}}_{p_1})1\/r}...p_g^{(f^L_{p_g}-f^{K_\\mathcal{I}}_{p_g})1\/r})}$ under the canonical embedding of ${L}.$\n\\end{theorem}\n\\begin{proof}\n\\noindent Consider the following diagram:\n\\begin{center}\n\\begin{tikzcd}\n\\mathcal{O}_L \\arrow[r, hook] & L \\arrow[r, \"\\Sigma_L\"] & \\mathbb{C}^N\\\\\n\\mathcal{O}_{K_I} \\arrow[u, hook] \\arrow[r, hook] & K_I \\arrow[u, hook] \\arrow[r, \"\\Sigma_{K_I}\"] & \\mathbb{C}^{[K_I:\\mathbb{Q}]} \\arrow[u, \"\\beta^\\prime\"]\n\\end{tikzcd}\n\\end{center}\n Here $\\beta^\\prime$ is chosen to make the diagram commute. Each embedding of $K_I$ extends to $N\/[K_I:\\mathbb{Q}]$ embeddings of $L$. Then $\\beta^\\prime$ simply repeats the coordinates of $\\Sigma_{K_I}$ $N\/r$ times, for $r = [K_I:\\mathbb{Q}]$, by the definition of $K_I$. So $\\|\\beta^\\prime(x)\\| = \\sqrt{N\/r}\\|x\\|$, for any $x\\in \\Sigma_{K_I}(K_I)$. Set $\\mathfrak{c} = I\\cap\\mathcal{O}_{K_I}$. Then $det(\\mathfrak{c}) = \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\sqrt{|disc(K_I\/\\mathbb{Q})|}$. So Hermite-SVP solution $v\\in\\mathfrak{c}$ satisfies $\\|v\\|\\leq\\gamma(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\sqrt{|disc(K_I\/\\mathbb{Q})|})^{1\/r}$. Also note ${disc}({L}\/\\mathbb{Q})={disc}({K_I}\/\\mathbb{Q})^{N \/ r} \\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))$. Then \n \\begin{align*}\n \\|\\beta^\\prime(v)\\|&\\leq\\sqrt{N\/r}\\|v\\|\\leq\\sqrt{N\/r}\\gamma\\cdot(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\sqrt{|disc(K_I\/\\mathbb{Q})|})^{1\/r}\\\\\n &= \\sqrt{N\/r}\\gamma\\cdot \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{1\/r}\\big(\\frac{disc(L\/\\mathbb{Q})^{r\/N}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{r\/N}}\\big)^{1\/2r}\\\\\n &= \\sqrt{N\/r}\\gamma\\cdot \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{1\/r}\\frac{disc(L\/\\mathbb{Q})^{1\/2N}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}\\\\\n &= \\gamma\\frac{\\sqrt{N\/r}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}\\cdot \\big(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{N\/r}\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}.\n \\end{align*}\n The norm is multiplicative: $\\text{Norm}_{K_I\/\\mathbb{Q}}(I) = \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathcal{P}_1)...\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathcal{P}_g)$, where $\\mathcal{P}_i=\\mathfrak{p}_i\\cap K_I$. All the $\\mathfrak{p}_i$ lie above unramified primes $p_i$, so we can write $e^{L}_{p_i}=1$. Moreover, we have $\\text{Norm}_{L\/\\mathbb{Q}}=\\text{Norm}_{K_I\/\\mathbb{Q}}\\circ \\text{Norm}_{L\/K_I}$. As a result, $\\text{Norm}_{L\/\\mathbb{Q}}(I) = p_1^{f^L_{p_1}}...p_g^{f^L_{p_g}}$. Also, $\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c}) = p_1^{f^{K_I}_{p_1}}...p_g^{f^{K_I}_{p_g}}$. Thus $\\text{Norm}_{L\/\\mathbb{Q}}(I) = p_1^{f^L_{p_1}}...p_g^{f^L_{p_g}} = (p_1^{f^{K_I}_{p_1}}...p_g^{f^{K_I}_{p_g}})\\cdot(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_1}}) = \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\cdot(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_g}})$, so we can rewrite\n \\begin{align*}\n \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c}) = \\text{Norm}_{L\/\\mathbb{Q}}(I)\/(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_g}}).\n \\end{align*} \n Then we have, setting $\\gamma^\\prime = \\gamma\\frac{\\sqrt{N\/r}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}$,\n\\begin{align*}\n\\|\\beta^\\prime(v)\\|&\\leq \\gamma\\frac{\\sqrt{N\/r}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}\\cdot \\big(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{N\/r}\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}\\\\\n&= \\gamma^\\prime\\cdot \\big(\\big(\\text{Norm}_{L\/\\mathbb{Q}}(I)\/(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_g}})\\big)^{N\/r}\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}\\\\\n&= \\gamma^\\prime\\cdot \\big(\\text{Norm}_{L\/\\mathbb{Q}}(I)^{N\/r}\/(p_1^{(f^L_{p_1}-f^{K_I}_{p_1})N\/r}...p_g^{(f^L_{p_g}-f^{K_I}_{p_g})N\/r})\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}\\\\\n&= \\gamma^\\prime\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(I)^{1\/r-1\/N}}{(p_1^{(f^L_{p_1}-f^{K_I}_{p_1})1\/r}...p_g^{(f^L_{p_g}-f^{K_I}_{p_g})1\/r})}\\cdot \\big(\\text{Norm}_{L\/\\mathbb{Q}}(I)\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N},\n\\end{align*}\nas required.\n \\end{proof}\n \\noindent Note that when $\\mathcal{I} = \\mathfrak{p}$, $K_I$ is the regular decomposition field, and $f^{K_I}_p = 1$ and $f^{L}_p = N\/r$, so $\\text{Norm}_{L\/\\mathbb{Q}}(I)^{1\/r-1\/N}=p^{N\/r(1\/r-1\/N)} = p^{N\/r^2-1\/r}$ and $p^{(f^L_{p}-f^{K_I}_{p})1\/r}) = p^{N\/r^2-1\/r}$, and we have recovered the result of the original paper.\n \\subsection{Solving Hermite-SVP for module lattices defined over a Galois extension}\nThe above method can be extended to the case of $\\mathcal{O}_L$-modules. As before, $L$ is a field that is Galois over $\\mathbb{Q}$ with ring of integers $\\mathcal{O}_L$. Suppose $\\mathcal{I}_1,\\dots, \\mathcal{I}_d \\subseteq \\mathcal{O}_L$ are some ideals of $\\mathcal{O}_L$ and $\\mathbf{b}_1,\\dots, \\mathbf{b}_d \\in L^D$ for some integer $d \\leq D$ that are linearly independent over $L$ (that is, none of the vectors can be expressed as a linear sum of the others over $L$). We define an $\\mathcal{O}_L$-module $M$ with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle$ as the direct sum\n\\begin{align*}\n    M=\\bigoplus_{k=1}^d \\mathcal{I}_k \\mathbf{b}_k,\n\\end{align*}\nwhich is a $\\mathbb{Z}$-module of dimension $d[L:\\mathbb{Q}]$. We define the volume of $M$ by\n\\begin{align*}\n    \\text{Vol}(M)=|\\text{disc}(L\/ \\mathbb{Q})|^d\\text{Norm}_{L\/\\mathbb{Q}}(\\det(B^\\dagger B))\\prod_{k=1}^d \\text{Norm}_{L\/\\mathbb{Q}}(I_k)^2,\n\\end{align*}\nwhere $B$ is the matrix composed of the columns $\\mathbf{b}_1,\\dots, \\mathbf{b}_d$ and $\\dagger$ denotes the Hermitian transpose of a matrix. Then, for any $\\mathbf{v}=(v_1,\\dots,v_D) \\in M$, the lattice $\\mathcal{L}_M$ generated by $M$ under the canonical embedding is the lattice whose elements take the form $(\\sigma_1(v_1),\\dots,\\sigma_{1}(v_D),\\sigma_2(v_1),\\dots,\\sigma_2(v_D),\\dots,\\sigma_{[L:\\mathbb{Q}]}(v_D))$. We say a vector $\\mathbf{v}$ is a solution to the $\\gamma$-Hermite-SVP in $M$ if it satisfies $\\|\\mathbf{v}\\| \\leq \\gamma \\text{Vol}(M)^{1\/2d[L:\\mathbb{Q}]}$. By abuse of notation, we say that for any $\\mathbf{v}=(v_1,\\dots,v_D) \\in L^D$, $\\sigma(\\mathbf{v})=(\\sigma(v_1),\\dots,\\sigma(v_D))$ for all $\\sigma \\in \\text{Gal}(L\/\\mathbb{Q})$ and \\\\$\\text{Trace}_{L\/K}(\\mathbf{v})=\\sum_{\\sigma \\in \\text{Gal}(L\/K)}\\sigma(\\mathbf{v})$ for any subfield $K$ of $L$. We call the module $\\mathcal{I}_k \\mathbf{b}_k$ a \\emph{pseudo-ideal} for each $k$, and define the decomposition group $\\Delta_k=\\{\\sigma \\in \\text{Gal}(L\/\\mathbb{Q}): \\sigma(\\mathcal{I}_k\\mathbf{b}_k)=\\mathcal{I}_k\\mathbf{b}_k\\}$.\n\\begin{lemma}\\label{separate}\nLet $\\mathcal{I}_k\\mathbf{b}_k$ be a pseudo-ideal in $L^D$, and let $\\Delta_k$ denote the decomposition group of $\\mathcal{I}_k\\mathbf{b}_k$ and $K_k$ the decomposition field of $\\Delta_k$. Then there exists an $\\alpha_k \\in L$ such that $\\mathbf{b}_k=\\alpha_k\\mathbf{b}_k^{\\prime}$, where $\\mathbf{b}_k^{\\prime} \\in K_k^D$.\n\\end{lemma}\n\\begin{proof}\nLet $x_k$ be some element of $\\mathcal{I}_k$. By the definition of the decomposition group, for every $\\sigma \\in \\Delta_i$ we must have $\\sigma(x_k\\mathbf{b}_k) \\in \\mathcal{I}_k \\mathbf{b}_k$ and so there exists some $x_{\\sigma,k} \\in \\mathcal{I}_k$ such that $x_{\\sigma,k}\\mathbf{b}_k=\\sigma(x_k\\mathbf{b}_k)$. Then, if we take $x_k$ to be some element of $\\mathcal{O}_{K_k}$, we have\n\\begin{align*}\n    x_k\\text{Trace}_{L\/K}(\\mathbf{b}_k)=\\text{Trace}_{L\/K}(x_k\\mathbf{b}_k)=\\left(\\sum_{\\sigma \\in \\Delta_k}x_{k,\\sigma}\\right)\\mathbf{b}_k \\in K_k^D,\n\\end{align*}\nand so setting $\\mathbf{b}_k^{\\prime}=\\left(\\sum_{\\sigma \\in \\Delta_k}x_{k,\\sigma}\\right)\\mathbf{b}_k$, $\\alpha_k=\\left(\\sum_{\\sigma \\in \\Delta_k}x_{k,\\sigma}\\right)^{-1}$, the lemma holds.\n\\end{proof}\nUsing this lemma, if we have a module equivalent to the direct sum of the pseudo-ideals $\\mathcal{I}_k\\mathbf{b}_k$ with decomposition groups $\\Delta_k$ and decomposition fields $K_k$, then we may represent the module as\n\\begin{align*}\n    M=\\bigoplus_{k=1}^d \\alpha_k\\mathcal{I}_k\\mathbf{b}_k^\\prime,\n\\end{align*}\nwhere $\\mathbf{b}_k^\\prime \\in K_k^D$.\n\\begin{theorem}\nLet $M$ be a module with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle_{k=1}^d $, and suppose that each pseudo-ideal $\\mathcal{I}_k\\mathbf{b}_k$ has decomposition group $\\Delta_k$ and decomposition field $K_k$. Denote by $\\langle \\mathcal{I}_k \\alpha_k, \\mathbf{b}_k^{\\prime} \\rangle_{k=1}^d$ the pseudo-basis that also represents $M$ such that $\\mathbf{b}_k^{\\prime} \\in K_k^D$. Let $\\mathcal{J}_k=Q_k\\alpha_k\\mathcal{I}_k$ where $Q_k$ is a rational integer such that $\\mathcal{J}_k$ is an ideal of $\\mathcal{O}_L$ and $\\mathcal{J}_k=\\prod_{i=1}^{g_k}\\mathfrak{p}_i^{(k)}$ where $\\mathfrak{p}_i^{(k)}$ are prime ideals lying above an unramified rational prime $p_i^{(k)}$ with inertial degree $f_{p_i^{(k)}}^L$ in $\\mathcal{O}_L$ and $f_{p_i^{(k)}}^{K_k}$ in $\\mathcal{O}_{K_k}$. We let $\\mathfrak{c}_k=\\mathcal{J}_k \\cap \\mathcal{O}_{K_k}$ and let $\\mathcal{M}=\\bigoplus_{k=1}^d \\frac{1}{Q_k}\\mathfrak{c}_k\\mathbf{b}_k^{\\prime}$. If $K$ is the compositum of all $K_k$, $1 \\leq k \\leq d$, then a solution to $\\gamma$-Hermite-SVP under the canonical embedding yields a solution to $\\gamma^\\prime$-Hermite-SVP in $M$, where\n\\begin{align*}\n    \\gamma^{\\prime}=\\gamma \\frac{\\sqrt{[L:K]}}{\\sqrt{|\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))}^{\\frac{1}{[L:\\mathbb{Q}]}}}\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d[K:\\mathbb{Q}]}-\\frac{1}{d[L:\\mathbb{Q}]}}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{1}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}.\n\\end{align*}\n\\end{theorem}\n\\begin{proof}\n\\noindent Consider the following diagram:\n\\begin{center}\n\\begin{tikzcd}\nM \\arrow[r, hook] & L^D \\arrow[r, \"\\Sigma_{L^D}\"] & \\mathbb{C}^{d[L:\\mathbb{Q}]}\\\\\n\\mathcal{M} \\arrow[u, hook] \\arrow[r, hook] & K^D \\arrow[u, hook] \\arrow[r, \"\\Sigma_{K^D}\"] & \\mathbb{C}^{d[K:\\mathbb{Q}]} \\arrow[u, \"\\beta^\\prime\"]\n\\end{tikzcd}\n\\end{center}\nHere, $\\beta^\\prime$ is chosen so that the diagram commutes. Each embedding of $K$ extends to $[L:K]$ embeddings of $L$, so $\\beta^{\\prime}$ repeats the coordinates of $\\Sigma_{K^D}$ $[L:K]$ times by the definition of $K$ being the compositum of fixed fields, so $\\|\\beta^{\\prime}(\\mathbf{v})\\|=\\sqrt{[L:K]}\\|\\mathbf{v}\\|$ for any $\\mathbf{v} \\in \\Sigma_{K^D}(K)$. Since the norm is multiplicative, $\\text{Norm}_{L\/\\mathbb{Q}}=\\text{Norm}_{K\/\\mathbb{Q}} \\circ \\text{Norm}_{L\/K}$. Now, we have\n\\begin{align*}\n\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)=\\prod_{i=1}^{g_k}{p_i^{(k)}}^{f_{p_i^{(k)}}^L}\n\\end{align*} \nand \n\\begin{align*}\n\\text{Norm}_{K\/\\mathbb{Q}}(\\mathfrak{c}_k)=\\left(\\text{Norm}_{K_k\/\\mathbb{Q}}(\\mathfrak{c}_k)\\right)^{[K:K_k]}=\\prod_{i=1}^{g_k}{p_i^{(k)}}^{[K:K_k]f_{p_i^{(k)}}^{K_k}},\n\\end{align*}\nand so we have the representation\n\\begin{align*}\n\\text{Norm}_{K\/\\mathbb{Q}}(\\mathfrak{c}_k)=\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}}}.\n\\end{align*}\nTherefore,\n\\begin{align*}\n    &\\text{Vol}(\\mathcal{M})=|\\text{disc}(K\/\\mathbb{Q})|^d\\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)\\prod_{k=1}^d\\text{Norm}_{K\/\\mathbb{Q}}(\\mathfrak{c}_k)^2\\prod_{k=1}^dQ_k^{-2[K:\\mathbb{Q}]}\n    \\\\&=|\\text{disc}(K\/\\mathbb{Q})|^d\\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^2}{Q_k^{2[K:\\mathbb{Q}]}\\prod_{i=1}^{g_k}{p_i^{(k)}}^{2\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}},\n\\end{align*}\nwhere $B$ is the matrix made up composed of the vectors $\\mathbf{b}_1^{\\prime},\\dots, \\mathbf{b}_d^{\\prime}$, and using the fact that $\\text{disc}(L\/\\mathbb{Q})=\\text{disc}(K\/\\mathbb{Q})^{[L:K]}\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))$,\n\\begin{align*}\n    &\\text{Vol}(M)=|\\text{disc}(L\/\\mathbb{Q})|^d\\text{Norm}_{L\/\\mathbb{Q}}(B^{\\dagger}B)\\prod_{k=1}^d\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^2\\prod_{k=1}^dQ_k^{-2[L:\\mathbb{Q}]}\n    \\\\&=|\\text{disc}(K\/\\mathbb{Q})|^{d[L:K]}\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))^d\\\\&\\cdot \\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)^{[L:K]}\\prod_{k=1}^d\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^2\\prod_{k=1}^dQ_k^{-2[L:\\mathbb{Q}]}.\n\\end{align*}\nHence, if we have a solution $\\mathbf{v} \\in \\mathcal{M}$ for the Hermite-SVP with factor $\\gamma$, then\n\\begin{align*}\n    &\\|\\beta(\\mathbf{v})\\|^2=[L:K]\\|\\mathbf{v}\\|^2\\leq \\gamma^2 [L:K] \\text{Vol}(\\mathcal{M})^{1\/d[K:\\mathbb{Q}]}\n    \\\\&= \\gamma^2 [L:K]|\\text{disc}(K\/\\mathbb{Q})|^{1\/[K:\\mathbb{Q}]}\\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)^{1\/d[K:\\mathbb{Q}]}\\\\ &\\cdot\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{2\/d[K:\\mathbb{Q}]}}{Q_k^{2\/d}\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{2}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}\n    \\\\&=\\frac{\\gamma^2[L:K]}{|\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))|^{1\/[L:\\mathbb{Q}]}}\\text{Vol}(M)^{1\/d[L:\\mathbb{Q}]}\\\\& \\cdot\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{2}{d}\\left(\\frac{1}{[K:\\mathbb{Q}]}-\\frac{1}{[L:\\mathbb{Q}]}\\right)}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{2}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}},\n\\end{align*}\nas required.\n\\end{proof}\nThough the factor we have obtained for general ideal lattices and module lattices may seem somewhat convoluted and tricky to interpret, we may make two remarks. The first, if $p_i^{(k)}\\mathcal{O}_{K_k}=\\prod_{j=1}^{t_{i,k}}\\mathfrak{p}_{i,j}^{(k)}$ where $\\mathfrak{p}_{i,j}^{(k)}$ are prime ideals of $\\mathcal{O}_{K_k}$ and each $\\mathfrak{p}_{i,j}^{(k)}$ is inert in $L$ for all $i,j,k$, then $\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d}\\left(\\frac{1}{[K:\\mathbb{Q}]}-\\frac{1}{[L:\\mathbb{Q}]}\\right)}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{1}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}=1$ (set $d=1$ for the ideal lattice case). Secondly, we may attain an upper bound that is easier to comprehend:\n\\begin{align*}\n    &\\gamma \\frac{\\sqrt{[L:K]}}{\\sqrt{|\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))}^{\\frac{1}{[L:\\mathbb{Q}]}}}\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d[K:\\mathbb{Q}]}-\\frac{1}{d[L:\\mathbb{Q}]}}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{1}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}\\\\\n    &\\leq \\gamma \\sqrt{[L:K]} \\prod_{k=1}^d \\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d[K:\\mathbb{Q}]}-\\frac{1}{d[L:\\mathbb{Q}]}},\n\\end{align*}\nand this value can be determined without having to know the prime decomposition of the ideals.\n\\section{Prime Ideals of Cyclotomic Fields}\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{s2^{n+1}})$}\nWe let $s$ be some positive odd integer, $s \\geq 3$. The following is Theorem 2.2 from \\cite{wangwang}. \n\\begin{theorem}\\label{s2factorisation}\nLet $q$ be an odd prime power, and let $s \\geq 3$ be any odd number such that $\\gcd(q,s)=1$, and let $q^{\\phi(s)}=m2^A+1$ for some odd $m$, $A \\geq 1$. Then, for any $A -1\\leq n$ and for any irreducible factor $f(x)$ of $\\Phi_{s2^A}(x)$ over $\\mathbb{F}_q$, then $f(x^{2^{n-A+1}})$ is also irreducible over $\\mathbb{F}_q$. Moreover, all irreducible factors of $\\Phi_{s2^{n+1}}(x)$ are obtained in this way.\n\\end{theorem}\n\\begin{theorem}\\label{s2svp}\nFor any prime ideal $\\mathfrak{p}=\\langle \\rho,f(\\zeta_{s2^{n+1}}) \\rangle$ of $\\mathcal{O}_L$ for some rational prime $\\rho$, $\\gcd(\\rho,s)=\\gcd(\\rho,2)=1$ and irreducible polynomial $f(x)$ of $\\Phi_{s2^{n+1}}$ in $\\mathbb{F}_\\rho[x]$, write $\\rho^{\\phi(s)}=m2^A+1$ where $m$ is an odd integer and $A \\geq 1$, and let $r=\\min\\{A-1,n\\}$. Then, given an oracle that can solve SVP for $\\phi(s2^{r+1})$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in \\\\$\\text{poly}(\\phi(s2^{n+1}),\\log_2\\rho)$ time with the canonical embedding.\n\\end{theorem}\n\\begin{proof}\nWe assume that $n \\geq A$ otherwise the theorem is vacuously true, so $r=A-1$. Let\n\\begin{align*}\n    G=\\{\\sigma_i: \\gcd(i,2)=\\gcd(i,s)=1\\}\n\\end{align*}\ndenote the Galois group of $L$ over $\\mathbb{Q}$, where\n\\begin{align*}\n    &\\sigma_i: \\mathbb{Q}(\\zeta_{s2^{n+1}}) \\to  \\mathbb{Q}(\\zeta_{s2^{n+1}}),\\\\\n    &\\sigma_i(\\zeta_{s2^{n+1}}^k)=\\zeta_{s2^{n+1}}^{ki}.\n\\end{align*}\nBy Theorem \\ref{s2factorisation}, for any factor $f(x)$ of $\\Phi_{s2^{n+1}}(x)$ that is irreducible in $\\mathbb{F}_\\rho[x]$, there exists a polynomial $g(x)$ that is a factor of $\\Phi_{s2^{r+1}}(x)$ that is irreducible over $\\mathbb{F}_{\\rho}[x]$ such that $f(x)=g(x^{2^{n-r}})$. Then the prime ideal lattice $\\mathfrak{p}$ can be represented by\n\\begin{align*}\n    \\langle \\rho,f(\\zeta_{s2^{n+1}}) \\rangle = \\langle \\rho,g(\\zeta_{s2^{r+1}}) \\rangle.\n\\end{align*}\nFor any $1 \\leq k \\leq 2^{n-r}-1$, the map $\\sigma_{ks2^{r+1}+1}$ fixes $\\zeta_{s2^{n+1}}^{l2^{n-r}}$ for any integer $0 \\leq l <s2^{r+1}$. Moreover, since $\\gcd(ks2^{r+1}+1,2)=\\gcd(ks2^{r+1},s)=1$, each subset $H_k$ of $G$ generated $\\sigma_{ks2^{r+1}+1}$ forms a cyclic group, and so the set $H=H_1 \\times H_2 \\times \\dots \\times H_{2^{n-r}-1}$ forms a subgroup of the decomposition group of $\\mathfrak{p}$, since both $\\rho$ and $f(\\zeta_{s2^{n+1}})=g(\\zeta_{2^{n+1}}^{2^{n-r}})$ are fixed by each $\\sigma_i \\in H$. $K=\\mathbb{Q}(\\zeta_{s2^{n+1}}^{2^{n-r}})$ must be the fixed field of the group $H$, as for all $i \\in \\left(\\mathbb{Z}\/s2^{n+1}\\mathbb{Z}\\right)^{\\times}$,\n\\begin{align*}\n    \\sigma_i(\\zeta_{s2^{n+1}}^{2^{n-r}})=\\zeta_{s2^{n+1}}^{2^{n-r}} \\iff i \\equiv 1 \\mod s2^{r+1}.\n\\end{align*}\nNote that $\\mathcal{O}_K$ has the $\\mathbb{Z}$-basis $\\{1,\\zeta_{s2^{n+1}}^{2^{n-r}},\\zeta_{s2^{n+1}}^{2(2^{n-r})}, \\dots, \\zeta_{s2^{n+1}}^{(\\phi(s2^{r+1})-1)(2^{n-r})}\\}$. Letting $\\mathfrak{c}=\\mathfrak{p} \\cap \\mathcal{O}_K$, we claim that\n\\begin{align*}\n    \\mathfrak{p}=\\bigoplus_{k=0}^{2^{n-r}}\\zeta_{s2^{n+1}}^k\\mathfrak{c}.\n\\end{align*}\nFor any $a \\in \\mathfrak{p}$, there exist integers $z_i,w_i$ such that\n\\begin{align*}\n    a&=\\sum_{i=0}^{\\phi(s2^{n+1})-1}z_i\\zeta_{s2^{n+1}}^if(\\zeta_{s2^{n+1}})+\\sum_{i=0}^{\\phi(s2^{n+1})-1}w_i\\zeta_{s2^{n+1}}\\rho\n    \\\\&=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^k\\sum_{j=0}^{\\phi(s2^{r+1})-1}\\left(z_{k+j2^{n-r}}\\zeta_{s2^{n+1}}^{j2^{n-r}}f(\\zeta_{s2^{n-r}})+w_{k+j2^{n-r}}\\zeta_{s2^{n+1}}^{j2^{n-r}}\\rho\\right)\n    \\\\&=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^k\\Bigg(\\left(\\sum_{j=0}^{\\phi(s2^{r+1})-1}z_{k+j2^{n-r}}\\zeta_{s2^{n+1}}^{j2^{n-r}}\\right)f(\\zeta_{s2^{n+1}})\\\\&+\\left(\\sum_{j=0}^{\\phi(s2^{r+1}-1}w_{k+j2^{n-r}}\\zeta_{s2^{n+1}}^{j2^{n-r}}\\right)\\rho\\Bigg),\n\\end{align*}\nwhich proves our claim. Now, for any $x_k \\in \\mathfrak{c}, 0 \\leq k \\leq 2^{n-r}-1$, let $x=\\sum_{k=0}^{2^{n-r}-1}x_k \\zeta_{s2^{n+1}}^k \\in \\mathfrak{p}$. Then the quadratic form induced by the ideal lattice $\\mathfrak{p}$ is given by\n\\begin{align*}\n    &\\text{Trace}_{L\/\\mathbb{Q}}(x\\overline{x})=\\text{Trace}_{L\/\\mathbb{Q}}\\left(\\sum_{k,l=0}^{2^{n-r}-1}x_k\\overline{x_l}\\zeta_{s2^{n+1}}^{k-l}\\right)\\\\&=\\sum_{i=0: \\gcd(i,s)=\\gcd(i,2)=1}^{s2^{n+1}-1}\\sum_{k,l=0}^{2^{n-r}-1}\\sigma_i\\left(x_k\\overline{x_l}\\zeta_{s2^{n+1}}^{k-l}\\right)\n    \\\\&=\\sum_{i=0: \\gcd(i,s)=\\gcd(i,2)=1}^{s2^{r+1}-1}\\sum_{j=0}^{2^{n-r}-1}\\sum_{k,l=0}^{2^{n-r}-1}\\sigma_{i+js2^{r+1}}(x_k\\overline{x_l})\\zeta_{s2^{n+1}}^{(i+js2^{r+1})(k-l)}\n    \\\\&=\\sum_{i=0: \\gcd(i,s)=\\gcd(i,2)=1}^{s2^{r+1}-1}\\sum_{j=0}^{2^{n-r}-1}\\sum_{k,l=0}^{2^{n-r}-1}\\sigma_i(x_k\\overline{x_l})\\zeta_{s2^{n+1}}^{(i+js2^{r+1})(k-l)},\n\\end{align*}\nbut note that we have\n\\begin{align*}\n    \\sum_{j=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^{(i+js2^{r+1})(k-l)}=\\sum_{j=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^{i(k-l)}\\zeta_{2^{n-r}}^{j(k-l)}=\n    \\begin{cases}\n    2^{n-r} \\hspace{2mm} &\\text{if} \\hspace{2mm} k=l,\n    \\\\ 0 \\hspace{2mm} &\\text{otherwise.}\n    \\end{cases}\n\\end{align*}\nHence\n\\begin{align*}\n    \\text{Trace}_{L\/\\mathbb{Q}}(x\\overline{x})&=2^{n-r}\\sum_{i=0: \\gcd(i,s)=\\gcd(i,2)=1}^{s2^{r+1}-1}\\sum_{k=0}^{2^{n-r}-1}\\sigma_i(x_k\\overline{x_k})\\\\&=2^{n-r}\\sum_{k=0}^{2^{n-r}-1}\\text{Trace}_{K\/\\mathbb{Q}}(x_k\\overline{x_k}),\n\\end{align*}\nand so $\\lambda_1(\\mathfrak{p})=\\lambda_1(\\mathfrak{c})$, as required. The algorithm below summarises how to find the shortest nonzero vector in a prime ideal lattice $\\mathfrak{p}$. The most time-consuming step in the algorithm below is Step 2, and all other steps may be performed in $\\text{poly}(\\phi(s2^{n+1}),\\log_2 \\rho)$ time.\n\\end{proof}\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{A prime ideal $\\mathfrak{p}=\\langle \\rho, f(\\zeta_{s2^{n+1}}) \\rangle$ in $\\mathbb{Z}[\\zeta_{s2^{n+1}}]$, where $\\rho$ is odd and $\\gcd(\\rho,s)=1$.} \\Output{A shortest vector in the corresponding prime ideal lattice.} \\BlankLine \n\\nl Compute the ideal $\\mathfrak{c}$ generated by $\\rho$ and $f(\\zeta_{s2^{n+1}})$ in $\\mathcal{O}_K$ where $K=\\mathbb{Q}(\\zeta_{s2^{n+1}}^{2^{n-r}})$.\n\\\\\n\\nl Find a shortest vector $v$ in the $\\phi(s2^{r+1})$-dimensional lattice $\\mathfrak{c}$.\n\\\\\n\\nl Output v.\n\\caption{SVP algorithm for prime ideal lattices of $\\mathbb{Z}[\\zeta_{s2^{n+1}}]$}\n\\end{algorithm}\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{sp^{n+1}})$}\nThe following theorem is a generalisation of Theorem 2 in \\cite{factorpn}.\n\\begin{theorem}\\label{spfactorisation}\nLet $s,p,q$ be positive integers such that $p$ is an odd prime, $q$ is a prime power and $\\gcd(s,p)=\\gcd(q,p)=1$. Suppose $q^{\\phi(s)} \\equiv a \\mod p$, for some integer $\\gcd(a,p)=1$ and set $q^{\\phi(s)}=mp^A+a$ for some integer $m$ such that $\\gcd(m,p)=1$ and some integer $A \\geq 0$. Then for any $n \\geq A-1$ and any irreducible factor $f(x)$ of the cyclotomic polynomial $\\Phi_{p^As}(x)$ over $\\mathbb{F}_q$, $f(x^{n-A+1})$ is also irreducible. Moreover, all the irreducible factors of $\\Phi_{p^{n+1}s}(x)$ are obtained in this way.\n\\end{theorem}\n\\begin{proof}\nSee Appendix A.\n\\end{proof}\n\\begin{theorem}\\label{spsvp}\nFor any prime ideal $\\mathfrak{p}=\\langle \\rho,f(\\zeta_{sp^{n+1}}) \\rangle$ where $\\rho$ is a positive rational prime, $\\gcd(\\rho,s)=\\gcd(\\rho,p)=1$ and irreducible polynomial $f(x)$ of the cyclotomic polynomial $\\Phi_{sp^{n+1}}(x)$ in $\\mathbb{F}_\\rho[x]$, assume that $\\rho^{\\phi(s)} \\equiv a \\mod p$ for some $\\gcd(a,p)=1$ and set $\\rho^{\\phi(s)}=mp^A + a$ for some positive integer $m$ such that $\\gcd(m,p)=1$ and some integer $A \\geq 1$ and let $r=\\min\\{A-1,n\\}$. Then, given an oracle that can solve SVP for $\\phi(sp^{r+1})$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in $\\text{poly}(\\phi(sp^{n+1}),\\log_2 \\rho)$ time with the canonical embedding.\n\\end{theorem}\n\\begin{proof}\nWe assume that $n \\geq A$ otherwise the theorem is vacuously true. Let\n\\begin{align*}\n    G=\\{\\sigma_i, 1 \\leq i \\leq sp^{n+1}-1: \\gcd(i,p)=\\gcd(i,s)=1\\}\n\\end{align*}\ndenote the Galois group of $L$ over $\\mathbb{Q}$, where\n\\begin{align*}\n    &\\sigma_i: \\mathbb{Q}(\\zeta_{sp^{n+1}}) \\to \\mathbb{Q}(\\zeta_{sp^{n+1}}),\\\\\n    &\\sigma_i(\\zeta_{sp^{n+1}}^k)=\\zeta_{sp^{n+1}}^{ki}.\n\\end{align*}\nBy Theorem \\ref{spfactorisation}, for any factor $f(x)$ of $\\Phi_{sp^{n+1}}(x)$ that is irreducible in $\\mathbb{F}_\\rho[x]$, there exists a polynomial $g(x)$ that is a factor of $\\Phi_{sp^{r+1}}(x)$ that is irreducible over $\\mathbb{F}_\\rho[x]$ such that $f(x)=g\\left(x^{p^{n-r}}\\right)$. Then the prime ideal $\\mathfrak{p}$ can be represented by\n\\begin{align*}\n    \\langle \\rho, f(\\zeta_{sp^{n+1}}) \\rangle = \\langle \\rho, g(\\zeta_{sp^{r+1}}) \\rangle.\n\\end{align*}\nFor any $1 \\leq k \\leq p^{n-r}-1$, the map $\\sigma_{ksp^{r+1}+1}$ fixes $\\zeta_{sp^{n+1}}^{lp^{n-r}}$ for any integer $0 \\leq l <sp^{r+1}$. Moreover, since $\\gcd(ksp^{r+1}+1,p)=\\gcd(ksp^{r+1},s)=1$, each subset $H_k$ of $G$ generated by $\\sigma_{ksp^{r+1}+1}$ forms a cyclic group, and so the set $H=H_1 \\times H_2 \\times \\dots \\times H_{p^{n-r}-1}$  forms a subgroup of the decompositiong group of $\\mathfrak{p}$, since both $\\rho$ and $f(\\zeta_{sp^{n+1}})=g\\left(\\zeta_{sp^{n+1}}^{p^{n-r}}\\right)$ are fixed by each $\\sigma_i \\in H$. Then $K=\\mathbb{Q}(\\zeta_{sp^{n+1}}^{p^{n-r}})$ must be the fixed field of the group $H$, as for all $i \\in \\left(\\mathbb{Z}\/sp^{n+1}\\mathbb{Z}\\right)^\\times$,\n\\begin{align*}\n    \\sigma_i(\\zeta_{sp^{n+1}}^{p^{n-r}})=\\zeta_{sp^{n+1}}^{p^{n-r}} \\iff i \\equiv 1 \\mod sp^{r+1}.\n\\end{align*}\nNote that $\\mathcal{O}_K$ has the $\\mathbb{Z}$-basis $\\{1,\\zeta_{sp^{n+1}}^{p^{n-r}},\\zeta_{sp^{n+1}}^{2p^{n-r}},\\dots, \\zeta_{sp^{n+1}}^{(\\phi(sp^{r+1})-1)p^{n-r}}\\}$. Letting $\\mathfrak{c}=\\mathfrak{p} \\cap \\mathcal{O}_K$, we claim that\n\\begin{align*}\n    \\mathfrak{p}=\\bigoplus_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k \\mathfrak{c}.\n\\end{align*}\nFor any $a \\in \\mathfrak{p}$, there exist integers $z_i,w_i$ such that\n\\begin{align*}\n    a&=\\sum_{i=0}^{\\phi(sp^{n+1})-1}z_i\\zeta_{sp^{n+1}}^if(\\zeta_{sp^{n+1}})+\\sum_{i=0}^{\\phi(sp^{n+1})-1}w_i\\zeta_{sp^{n+1}}\\rho\n    \\\\&=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k\\sum_{j=0}^{\\phi(sp^{r+1})-1}\\left(z_{k+jp^{n-r}}\\zeta_{sp^{n+1}}^{jp^{n-r}}f(\\zeta_{sp^{n-r}})+w_{k+jp^{n-r}}\\zeta_{sp^{n+1}}^{jp^{n-r}}\\rho\\right)\n    \\\\&=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k\\Bigg(\\left(\\sum_{j=0}^{\\phi(sp^{r+1})-1}z_{k+jp^{n-r}}\\zeta_{sp^{n+1}}^{jp^{n-r}}\\right)f(\\zeta_{sp^{n+1}})\\\\&+\\left(\\sum_{j=0}^{\\phi(sp^{r+1})-1}w_{k+jp^{n-r}}\\zeta_{sp^{n+1}}^{jp^{n-r}}\\right)\\rho\\Bigg),\n\\end{align*}\nwhich proves our claim. Now, for any $x_k \\in \\mathfrak{c}, 0 \\leq k \\leq p^{n-r}-1$, let $x=\\sum_{k=0}^{p^{n-r}-1}x_k\\zeta_{sp^{n+1}}^k \\in \\mathfrak{p}$. Then the quadratic form induced by the ideal lattice $\\mathfrak{p}$ is given by\n\\begin{align*}\n    &\\text{Trace}_{L\/\\mathbb{Q}}(x\\overline{x})=\\text{Trace}_{L\/\\mathbb{Q}}\\left(\\sum_{k,l=0}^{p^{n-r}-1}x_k\\overline{x_l}\\zeta_{sp^{n+1}}^{k-l}\\right)\\\\&=\\sum_{i=0: \\gcd(i,s)=\\gcd(i,p)=1}^{sp^{n+1}-1}\\sum_{k,l=0}^{p^{n-r}-1}\\sigma_i(x_k\\overline{x_l}\\zeta_{sp^{n+1}}^{k-l})\n    \\\\&=\\sum_{i=0: \\gcd(i,s)=\\gcd(i,p)=1}^{sp^{r+1}-1}\\sum_{j=0}^{p^{n-r}-1}\\sum_{k,l=0}^{p^{n-r}-1}\\sigma_{i+jsp^{r+1}}(x_k\\overline{x_l})\\zeta_{sp^{n+1}}^{(i+jsp^{r+1})(k-l)}\n    \\\\&=\\sum_{i=0: \\gcd(i,s)=\\gcd(i,p)=1}^{sp^{r+1}-1}\\sum_{j=0}^{p^{n-r}-1}\\sum_{k,l=0}^{p^{n-r}-1}\\sigma_i(x_k\\overline{x_l})\\zeta_{sp^{n+1}}^{(i+jsp^{r+1})(k-l)},\n\\end{align*}\nbut note that we have\n\\begin{align*}\n    \\sum_{j=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^{(i+jsp^{r+1})(k-l)}=\\sum_{j=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^{i(k-l)}\\zeta_{p^{n-r}}^{j(k-l)}=\n    \\begin{cases}\n    p^{n-r} \\hspace{2mm} &\\text{if} \\hspace{1mm} k=l,\\\\\n    0 \\hspace{2mm} &\\text{otherwise.}\n    \\end{cases}\n\\end{align*}\nHence\n\\begin{align*}\n    \\text{Trace}_{L\/\\mathbb{Q}}(x\\overline{x})&=p^{n-r}\\sum_{i=0: \\gcd(i,s)=\\gcd(i,p)=1}^{sp^{r+1}-1}\\sum_{k=0}^{p^{n-r}-1}\\sigma_i(x_k\\overline{x_k})\\\\&=p^{n-r}\\sum_{k=0}^{p^{n-r}-1}\\text{Trace}_{K\/\\mathbb{Q}}(x_k\\overline{x_k}),\n\\end{align*}\nand so $\\lambda_1(\\mathfrak{p})=\\lambda_1(\\mathfrak{c})$, as required. The algorithm below summarises how to find the shortest nonzero vector in a prime ideal lattice $\\mathfrak{p}$. The most time-consuming step in the algorithm below is Step 2, and all other steps may be performed in $\\text{poly}(\\phi(sp^{n+1}),\\log_2 \\rho)$ time.\n\\end{proof}\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{A prime ideal $\\mathfrak{p}=\\langle \\rho, f(\\zeta_{sp^{n+1}}) \\rangle$ in $\\mathbb{Z}[\\zeta_{sp^{n+1}}]$, where $\\gcd(p,\\rho)=\\gcd(\\rho,s)=1$ and $\\rho^{\\phi(s)} \\equiv \\pm 1 \\mod p$.} \\Output{A shortest vector in the corresponding prime ideal lattice.} \\BlankLine \n\\nl Compute the ideal $\\mathfrak{c}$ generated by $\\rho$ and $f(\\zeta_{sp^{n+1}})$ in $\\mathcal{O}_K$ where $K=\\mathbb{Q}(\\zeta_{sp^{n+1}}^{p^{n-r}})$.\n\\\\\n\\nl Find a shortest vector $v$ in the $\\phi(sp^{r+1})$-dimensional lattice $\\mathfrak{c}$.\n\\\\\n\\nl Output v.\n\\caption{SVP algorithm for prime ideal lattices of $\\mathbb{Z}[\\zeta_{sp^{n+1}}]$}\n\\end{algorithm}\n\\subsection{Some Special Prime Ideals of Cyclotomic Rings}\n\\begin{theorem}\nLet $L=\\mathbb{Q}(\\zeta_{s2^{n+1}})$ be a cyclotomic field for some positive odd integer $s \\geq 3$ and some integer $n \\geq 0$. Let $\\mathfrak{p}$ denote a prime ideal lying over a positive rational odd prime $\\rho$ such that $\\rho^{\\phi(s)} \\equiv 3 \\mod 4$. Then, given an oracle that can solve SVP for $\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in $\\text{poly}(\\phi(s2^{n+1}),\\log_2\\rho)$ time with the canonical embedding.\n\\end{theorem}\n\\begin{proof}\nFor some integer $N>1$, we must have $\\rho^{\\phi(s)} \\equiv 2l+1 \\mod 2^N$, and so for some integer $k$, we have $\\rho^{\\phi(s)} = 1+2l+2^Nk=1+2(2^{N-1}k+l)$. Since $2^{N-1}k+l$ is an odd integer and $N$ is taken totally arbitrarily, the claim holds by Theorem \\ref{s2svp}.\n\\end{proof}\n\\begin{theorem}\nLet $L=\\mathbb{Q}(\\zeta_{sp^{n+1}})$ be a cyclotomic field for some positive integer $s$ such that $\\gcd(s,p)=1$, an odd prime $p$ and some integer $n \\geq 0$. Let $\\mathfrak{p}$ denote a prime ideal lying over a positive rational odd prime $\\rho$ such that $\\rho^{\\phi(s)} =lp +a $ for some integers $l,a$, $\\gcd(l,p)=\\gcd(a,p)=1$. Then, given an oracle that can solve SVP for $(p-1)\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in $\\text{poly}(\\phi(sp^{n+1}),\\log_2 \\rho)$ time with the canonical embedding.\n\\end{theorem}\n\\begin{proof}\nFor some integer $N>1$, we must have $\\rho^{\\phi(s)} \\equiv lp +a \\mod p^N$, and so for some integer $k$, we have $\\rho^{\\phi(s)}=m a+pl+p^{N}k= a+p(p^{N-1}k+l)$. Since $\\gcd(p^{N-1}k+l,p)=1$ and $N$ is taken totally arbitrarily, the claim holds by Theorem \\ref{spsvp}.\n\\end{proof}\n\\section{General Ideals of Cyclotomic Rings}\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{s2^{n+1}})$}\nAs before, we set $s$ to be some odd integer greater than or equal to $3$.\n\\begin{theorem}\nLet $\\mathcal{I}$ be a nonzero ideal of $\\mathbb{Z}[\\zeta_{s2^{n+1}}]$ with prime factorisation\n\\begin{align*}\n    \\mathcal{I}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_t,\n\\end{align*}\nwhere $\\mathfrak{p}_i=(f_i(\\zeta_{s2^{n+1}}),\\rho_i)$ for rational primes $\\rho_i$ are (not necessarily distinct) prime ideals. If $\\rho_i$ is odd, write $\\rho_i^{\\phi(s)}=m_i2^{A_i}+1$, for some integer $m_i$ such that $\\gcd(m_i,2)=1$ and let $r=\\max\\{r_i\\}$, where\n\\begin{align*}\n    r_i=\n    \\begin{cases}\n    \\min\\{A_i-1,n\\}, \\hspace{0.5mm} &\\text{if} \\hspace{1mm} \\rho_i \\equiv 1 \\mod 2,\\\\\n    n \\hspace{0.5mm} &\\text{if} \\rho_i=2.\n    \\end{cases}\n\\end{align*}\nThen SVP in the lattice generated by $\\mathcal{I}$ can be solved via solving SVP in a $\\phi(s2^{r+1})$-dimensional lattice.\n\\end{theorem}\n\\begin{proof}\nIf $r=n$ the theorem vacuously holds, so we assume otherwise. We may assume WLOG that $r=r_1$. Following the notation of Theorem \\ref{s2svp}, we denote by\n\\begin{align*}\n    G=\\{\\sigma_i: 1 \\leq i \\leq s2^{n+1}-1, \\gcd(i,2)=\\gcd(i,s)=1\\}\n\\end{align*}\nthe Galois group of $L$, and consider the subgroup $H=H_1 \\times H_2 \\times \\dots \\times H_{2^{n-r-1}}$, where $H_k$ is the cyclic group generated by $\\langle \\sigma_{ks2^{r+1}+1}\\rangle$, which is a subgroup of the decomposition group of every $\\mathfrak{p}_i$, since $\\sigma_{ks2^{r+1}+1}(\\rho_i)=\\rho_i$, $\\sigma_{ks2^{r+1}+1}(f_i(\\zeta_{s2^{n+1}}))=\\sigma_{ks2^{r+1}+1}(g_i(\\zeta_{s2^{r+1}}))=g_i(\\zeta_{s2^{r+1}})=f_i(\\zeta_{s2^{n+1}})$, where $g_i(x)$ is an irreducible factor of $\\Phi_{s2^{A_i}}(x)$. As shown in Theorem \\ref{s2svp}, the fixed field of $H$ is $K=\\mathbb{Q}(\\zeta_{s2^{n+1}}^{2^{n-r}})$, which has ring of integers $\\mathcal{O}_K=\\mathbb{Z}[\\zeta_{s2^{n+1}}^{2^{n-r}}]$. Let $\\mathfrak{c}= \\mathcal{I} \\cap \\mathcal{O}_K$. We claim that for any $a \\in \\mathcal{I}$, there exist $a^{(k)} \\in \\mathfrak{c}$ for $0 \\leq k \\leq 2^{n-r}-1$ such that\n\\begin{align*}\n    a=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^ka^{(k)}.\n\\end{align*}\nWe prove the claim via induction. When $t=1$, the claim holds by Theorem \\ref{s2svp}, so we assume the claim holds for $t-1$. Letting $\\overline{\\mathcal{I}}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_{t-1}$, we have $\\mathcal{I}=\\mathfrak{p}_t\\mathcal{I}$. It suffices to show that for any $xy$, $x \\in \\overline{\\mathcal{I}}, y \\in \\mathfrak{p}_t$, there exist $b^{(k)} \\in \\mathcal{I} \\cap \\mathcal{O}_K$ for $0 \\leq k \\leq 2^{n-r}-1$ such that $xy=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^kb^{(k)}$. By the induction assumption, there exist $x^{(i)} \\in \\overline{\\mathcal{I}} \\cap \\mathcal{O}_K$, $y^{(j)} \\in \\mathfrak{p}_t \\cap \\mathcal{O}_K$, $0 \\leq i,j \\leq 2^{n-r}-1$ such that $x=\\sum_{i=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^ix^{(i)}$ and $y=\\sum_{j=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^jy^{(j)}$. Hence, we have\n\\begin{align*}\n    xy&=\\sum_{i,j=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^{i+j}x^{(i)}y^{(i)}\n    \\\\&=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=2^{n-r}}^{2\\cdot 2^{n-r}-2}\\zeta_{s2^{n+1}}^k\n    \\\\&=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=0}^{2^{n-r}-2}\\zeta_{s2^{n+1}}^k\\sum_{i+j=k+2^{n-r}}\\zeta_{s2^{n+1}}^{2^{n-r}}x^{(i)}y^{(j)}\n    \\\\&=\\sum_{k=0}^{2^{n-r}-2}\\zeta_{s2^{n+1}}^k\\left(\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+2^{n-r}}\\zeta_{s2^{n+1}}^{2^{n-r}}x^{(i)}y^{(j)}\\right)\\\\&+\\zeta_{s2^{n+1}}^{2^{n-r}-1}\\sum_{i+j=2^{n-r}-1}x^{(i)}y^{(j)}.\n\\end{align*}\nBy letting \\begin{align*}\nb^{(k)}=\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+2^{n-r}}\\zeta_{s2^{n+1}}^{2^{n-r}}x^{(i)}y^{(j)}\n\\end{align*} \nfor $0 \\leq k \\leq 2^{n-r}-2$ and \n\\begin{align*}\nb^{(2^{n-r}-1)}=\\sum_{i+j=2^{n-r}-1}x^{(i)}y^{(j)},\n\\end{align*} \nwe have proven our claim. As in Theorem \\ref{s2svp}, we have $\\lambda_1(\\mathcal{I})=\\lambda_1(\\mathfrak{c})$, as required.\n\\end{proof}\nThe following algorithm may be used to compute the shortest vector in $\\mathcal{I}$.\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{An ideal $\\mathcal{I}$.} \\Output{A shortest vector in the corresponding ideal lattice.} \\BlankLine \n\\nl \\For{$\\overline{r}=1$ \\KwTo $n$}{\n\\nl Compute a basis $(b^{(i)})_{0 \\leq i < \\phi(s2^{\\overline{r}+1})}$ of the ideal lattice $\\mathfrak{c}=\\mathcal{I} \\cap \\mathcal{O}_K$ where $K=\\mathbb{Q}(\\zeta_{s2^{n+1}}^{2^{n-\\overline{r}}})$. \\\\\n\\nl \\If{$(\\zeta_{s2^{n+1}}^jb^{(i)})_{0 \\leq i < \\phi(s2^{\\overline{r}+1}), 0 \\leq j \\leq 2^{n-\\overline{r}}}$ is exactly a basis of the ideal lattice $\\mathcal{I}$}{\n\\nl Find a shortest vector $v$ in the $\\phi(s2^{\\overline{r}+1})$-dimensional lattice $\\mathfrak{c}$.\\\\\n\\nl Output $v$.}\n}\n\\caption{SVP algorithm for general ideal lattices of $\\mathbb{Z}[\\zeta_{s2^{n+1}}]$}\n\\end{algorithm}\n\\\\\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{sp^{n+1}})$}\nAs before, $p$ is a positive, odd prime and $s$ is a positive integer such that $\\gcd(s,p)=1$.\n\\begin{theorem}\nLet $\\mathcal{I}$ be a nonzero ideal of $\\mathbb{Z}[\\zeta_{sp^{n+1}}]$ with prime factorisation\n\\begin{align*}\n    \\mathcal{I}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_t,\n\\end{align*}\nwhere $\\mathfrak{p}_i=(f_i(\\zeta_{sp^{n+1}}),\\rho_i)$ for rational primes $\\rho_i$ are (not necessarily distinct) prime ideals. If $\\rho_i^{\\phi(s)} \\equiv a \\mod p$ for some $\\gcd(p,a)=1$, write $\\rho_i^{\\phi(s)}=m_ip^{A_i}+1$ and let $r=\\max\\{r_i\\}$, where\n\\begin{align*}\n    r_i=\n    \\begin{cases}\n    \\min\\{A_i-1,n\\}, \\hspace{0.5mm} &\\text{if} \\hspace{1mm} \\rho_i^{\\phi(s)} \\equiv a \\mod p,\\\\\n    n \\hspace{0.5mm} &\\text{if $\\rho_i=p$}.\n    \\end{cases}\n\\end{align*}\nThen SVP in the lattice generated by $\\mathcal{I}$ can be solved via solving SVP in a $\\phi(sp^{r+1})$-dimensional lattice.\n\\end{theorem}\n\\begin{proof}\nIf $r=n$ the theorem vacuously holds, so we assume otherwise. We may assume WLOG that $r=r_1$. Following the notation of Theorem \\ref{spsvp}, we denote by\n\\begin{align*}\n    G=\\{\\sigma_i: 1 \\leq i \\leq sp^{n+1}-1, \\gcd(i,p)=\\gcd(i,s)=1\\}\n\\end{align*}\nthe Galois group of $L$, and consider the subgroup $H=H_1 \\times H_2 \\times \\dots \\times H_{p^{n-r-1}}$, where $H_k$ is the cyclic group generated by $\\langle \\sigma_{ksp^{r+1}+1}\\rangle$, which is a subgroup of the decomposition group of every $\\mathfrak{p}_i$, since $\\sigma_{ksp^{r+1}+1}(\\rho_i)=\\rho_i$, $\\sigma_{ksp^{r+1}+1}(f_i(\\zeta_{sp^{n+1}}))=\\sigma_{ksp^{r+1}+1}(g_i(\\zeta_{sp^{r+1}}))=g_i(\\zeta_{sp^{r+1}})=f_i(\\zeta_{sp^{n+1}})$, where $g_i(x)$ is an irreducible factor of $\\Phi_{sp^{A_i}}(x)$. As shown in Theorem \\ref{spsvp}, the fixed field of $H$ is $K=\\mathbb{Q}(\\zeta_{sp^{n+1}}^{p^{n-r}})$, which has ring of integers $\\mathcal{O}_K=\\mathbb{Z}[\\zeta_{sp^{n+1}}^{p^{n-r}}]$. Let $\\mathfrak{c}= \\mathcal{I} \\cap \\mathcal{O}_K$. We claim that for any $a \\in \\mathcal{I}$, there exist $a^{(k)} \\in \\mathfrak{c}$ for $0 \\leq k \\leq p^{n-r}-1$ such that\n\\begin{align*}\n    a=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^ka^{(k)}.\n\\end{align*}\nWe prove the claim via induction. When $t=1$, the claim holds by Theorem \\ref{spsvp}, so we assume the claim holds for $t-1$. Letting $\\overline{\\mathcal{I}}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_{t-1}$, we have $\\mathcal{I}=\\mathfrak{p}_t\\mathcal{I}$. It suffices to show that for any $xy$, $x \\in \\overline{\\mathcal{I}}, y \\in \\mathfrak{p}_t$, there exist $b^{(k)} \\in \\mathcal{I} \\cap \\mathcal{O}_K$ for $0 \\leq k \\leq p^{n-r}-1$ such that $xy=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^kb^{(k)}$. By the induction assumption, there exist $x^{(i)} \\in \\overline{\\mathcal{I}} \\cap \\mathcal{O}_K$, $y^{(j)} \\in \\mathfrak{p}_t \\cap \\mathcal{O}_K$, $0 \\leq i,j \\leq p^{n-r}-1$ such that $x=\\sum_{i=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^ix^{(i)}$ and $y=\\sum_{j=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^jy^{(j)}$. Hence, we have\n\\begin{align*}\n    xy&=\\sum_{i,j=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^{i+j}x^{(i)}y^{(i)}\n    \\\\&=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=p^{n-r}}^{2 p^{n-r}-2}\\zeta_{sp^{n+1}}^k\n    \\\\&=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=0}^{p^{n-r}-2}\\zeta_{sp^{n+1}}^k\\sum_{i+j=k+p^{n-r}}\\zeta_{sp^{n+1}}^{p^{n-r}}x^{(i)}y^{(j)}\n    \\\\&=\\sum_{k=0}^{p^{n-r}-2}\\zeta_{sp^{n+1}}^k\\left(\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+p^{n-r}}\\zeta_{sp^{n+1}}^{p^{n-r}}x^{(i)}y^{(j)}\\right)\\\\&+\\zeta_{sp^{n+1}}^{p^{n-r}-1}\\sum_{i+j=p^{n-r}-1}x^{(i)}y^{(j)}.\n\\end{align*}\nBy letting \\begin{align*}\nb^{(k)}=\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+p^{n-r}}\\zeta_{sp^{n+1}}^{p^{n-r}}x^{(i)}y^{(j)}\n\\end{align*} \nfor $0 \\leq k \\leq p^{n-r}-2$ and \n\\begin{align*}\nb^{(p^{n-r}-1)}=\\sum_{i+j=p^{n-r}-1}x^{(i)}y^{(j)},\n\\end{align*} \nwe have proven our claim. As in Theorem \\ref{spsvp}, we have $\\lambda_1(\\mathcal{I})=\\lambda_1(\\mathfrak{c})$, as required.\n\\end{proof}\nThe following algorithm may be used to compute the shortest vector in $\\mathcal{I}$.\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{An ideal $\\mathcal{I}$.} \\Output{A shortest vector in the corresponding ideal lattice.} \\BlankLine \n\\nl \\For{$\\overline{r}=1$ \\KwTo $n$}{\n\\nl Compute a basis $(b^{(i)})_{0 \\leq i < \\phi(sp^{\\overline{r}+1})}$ of the ideal lattice $\\mathfrak{c}=\\mathcal{I} \\cap \\mathcal{O}_K$ where $K=\\mathbb{Q}(\\zeta_{sp^{n+1}}^{2^{n-\\overline{r}}})$. \\\\\n\\nl \\If{$(\\zeta_{sp^{n+1}}^jb^{(i)})_{0 \\leq i < \\phi(sp^{\\overline{r}+1}), 0 \\leq j \\leq p^{n-\\overline{r}}}$ is exactly a basis of the ideal lattice $\\mathcal{I}$}{\n\\nl Find a shortest vector $v$ in the $\\phi(sp^{\\overline{r}+1})$-dimensional lattice $\\mathfrak{c}$.\\\\\n\\nl Output $v$.}\n}\n\\caption{SVP algorithm for general ideal lattices of $\\mathbb{Z}[\\zeta_{sp^{n+1}}]$}\n\\end{algorithm}\n\\section{Modules over Cyclotomic Rings}\nThroughout this section, we use the same notation as in section 3.1. We let $L=\\mathbb{Q}(\\zeta_N)$ be the cyclotomic field of conductor $N$, where $N$ is of the form $sq^{n+1}$ for some positive integer $s$ and some $q$ where $q$ is $2$ or an odd prime, $\\gcd(s,q)=1$. We take a module $M$ with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle_{k=1}^d$. As in section 3.1, we associate to each pseudo-ideal $\\mathfrak{I}_k\\mathbf{b}_k$ the decomposition group $\\Delta_{\\mathcal{I}_k}$ and the decomposition field $K_{\\mathcal{I}_k}$. As we showed in Lemma \\ref{separate}, we may alternatively represent the module $M$ using the pseudo-basis $\\langle \\alpha_k \\mathcal{I}_k,\\mathbf{b}_k^\\prime \\rangle_{k=1}^d$ where $\\mathbf{b}_k^\\prime \\in K_{\\mathcal{I}_k}^D$ and $\\alpha_k\\mathcal{I}_k$ are fractional ideals. Let $\\mathcal{J}_k=Q_k\\alpha_k\\mathcal{I}_k$ for some rational integer $Q_k$ such that $\\mathcal{J}_k$ is an ideal of $\\mathcal{O}_L$. Denote by $\\Delta_{\\mathcal{J}_k}$ the decomposition group of $\\mathcal{J}_k$ and by $r_k$ the maximum integer such that all the embeddings that fix $K_{k}=\\mathbb{Q}(\\zeta_{sq^{n+1}}^{q^{n-r_k}})$ also fix $K_{\\mathcal{I}_j}$ for $1 \\leq j \\leq d$ (that is, they must form a subgroup of $\\Delta_{\\mathcal{I}_j}$), and such that (as in section 5) we may express $\\mathcal{J}_k$ as\n\\begin{align*}\n    \\mathcal{J}_k=\\bigoplus_{i=0}^{q^{n-r_k}-1}\\mathfrak{c}_k\\zeta_{sq^{n+1}}^i,\n\\end{align*}\nwhere $\\mathfrak{c}_k=\\mathcal{J}_k \\cap \\mathcal{O}_{K_k}$. Then we may express $M$ by\n\\begin{align*}\n    M=\\bigoplus_{k=1}^d\\mathcal{I}_k\\mathbf{b}_k=\\bigoplus_{k=1}^d\\frac{1}{Q_k}\\mathbf{b}_k^\\prime\\bigoplus_{i=0}^{q^{n-r_k}-1}\\mathfrak{c}_k\\zeta_{sq^{n+1}}^i.\n\\end{align*}\nSince each $r_k$ are chosen so that the embeddings that fix the field $K_k$ also fix $K_{\\mathcal{I}_j}$ for $1 \\leq j \\leq d$ and under the canonical embedding of ideal lattices the coefficients next to the roots of unity in the expression above may be treated as orthogonal components (as we have already shown), under the canonical embedding of $M$, the terms multiplied by the roots of unity in the expression above may be treated as orthogonal components, and so the shortest vector of the submodule\n\\begin{align*}\n    \\mathcal{M}=\\bigoplus_{k=1}^d\\frac{1}{Q_k}\\mathfrak{c}_k\\mathbf{b}_k^\\prime\n\\end{align*}\nis also the shortest vector in the module $M$. This simplifies SVP in module lattices, as the original module generated a lattice of dimension $d\\phi(sq^{n+1})$ under the canonical embedding, whilst the submodule above generates a lattice of dimension at most $d\\phi(sq^{r+1})$ where $r=\\max_k\\{r_k\\} \\leq n$. The following algorithm details how to perform SVP in module lattices over cyclotomic rings, and uses notation and conventions that we have discussed in this section.\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{A module $M$ with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle_{k=1}^d$.} \\Output{A shortest vector in the corresponding module lattice.} \\BlankLine \n\\nl \\For{$1 \\leq i \\leq d$}{\n\\nl Compute the decomposition group $\\Delta_{\\mathcal{I}_i}$ of the pseudo-ideal $\\mathcal{I}_i\\mathbf{b}_i$ and decomposition field $K_{\\mathcal{I}_i}$. \\\\\n\\nl Set $x_i=\\text{Norm}_{L\/K_{\\mathcal{I}_i}}(\\mathcal{I}_i)$.\\\\\n\\nl \\For{$\\sigma \\in \\Delta_{\\mathcal{I}_i}$}{\n\\nl Find $x_{\\sigma,i} \\in \\mathcal{I}_i$ that satisfies $x_{\\sigma,i}\\mathbf{b}_i=\\sigma(x_i\\mathbf{b}_i)$}\n\\nl Set $\\alpha_i=\\left(\\sum_{\\sigma \\in \\Delta_{\\mathcal{I}_i}} x_{i,\\sigma}\\right)^{-1}$, $\\mathbf{b}_i^\\prime=\\alpha_i^{-1}\\mathbf{b}_i$\\\\\n\\nl Set $Q_i$ to be smallest integer such that $Q_i\\alpha_i\\mathcal{I}_i \\subseteq \\mathcal{O}_L$, set $\\mathcal{J}_i=Q_i\\alpha_i\\mathcal{I}_i$\n}\n\\nl Compute the field $K=\\mathbb{Q}(\\zeta_{sq^{n+1}}^{q^{n-t}})$ such that $K$ is a subfield of the compositum of all $K_{\\mathcal{I}_i}$, $1 \\leq i \\leq d$.\\\\\n\\nl \\For{$1 \\leq i \\leq d$}{\n\\For{$1 \\leq \\overline{r}_i \\leq n$}{\n\\nl Compute a basis $(b_i^{(j)})_{0 \\leq j \\leq \\phi(sq^{\\overline{r}_i+1})}$ of the ideal $\\mathfrak{c}_i=\\mathcal{J}_i \\cap \\mathcal{O}_{K_i}$ where $K_i=\\mathbb{Q}(\\zeta_{sq^{n+1}}^{q^{n-\\overline{r}_i}})$ \\\\\n\\If{$(\\zeta_{sq^{n+1}}^kb_i^{(j)})_{0 \\leq j \\leq \\phi(sq^{\\overline{r}_i+1}),0 \\leq k \\leq q^{n-\\overline{r}_i}}$ is exactly a basis of $\\mathcal{J}_i$ and $K_i \\subseteq K$}{\nSet $\\mathfrak{c}_i=\\mathcal{J}_i \\cap K_i$ and break\n}\n}\n}\n\\nl Find the shortest vector $\\mathbf{v}$ in the lattice generated by the module $\\mathcal{M}=\\bigoplus_{i=1}^d \\frac{1}{Q_i}\\mathfrak{c}_i \\mathbf{b}_i^{\\prime}$. \\\\\n\\nl Output $\\mathbf{v}$.\n\\caption{SVP algorithm for a module lattice $M$ over a cyclotomic ring.}\n\\end{algorithm}\n\\section{SVP Average-Case Hardness}\nWe fix a large $M$, and select a prime ideal uniformly randomly from the set\n\\begin{align*}\n    \\{\\mathfrak{p} \\hspace{2mm} \\text{a prime ideal}: N(\\mathfrak{p})<M\\}.\n\\end{align*}\nOur method of reduction to a $\\phi(s2^{r+1})$- or $\\phi(sp^{r+1})$-dimensional sublattice $\\mathfrak{c}$ only works when $p$ does not split completely in $L$, or equivalently, when $N(\\mathfrak{p})=\\rho$. By Chebotarev's density theorem, the number of primes less than $M$ that split completely in $L=\\mathbb{Q}(\\zeta_m)$ is approximately $\\frac{M}{\\phi(m)\\log(M)}$, and hence there are $\\frac{M}{\\log(M)}$ prime ideals lying above those primes for which our reduction method cannot be applied. Now, if our algorithm is to provide a reduction, we need $N(\\mathfrak{p})=\\rho^f<M$ for some positive integer $m$, and so the prime ideal must lie over a rational prime $\\rho$ such that $\\rho<\\sqrt{M}$. Hence, there are at most $\\sqrt{M}$ of such primes, and as such, there are at most:\n\\begin{itemize}\n    \\item $\\phi(s)2^{n-1}\\sqrt{M}$ when $L=\\mathbb{Q}(\\zeta_{s2^{n+1}})$ for odd, positive integer $s \\geq 3$,\n    \\item $\\phi(s)(p-1)p^{n-1}\\sqrt{M}$ when $L=\\mathbb{Q}(\\zeta_{sp^{n+1}})$ for odd positive integer $s$ and odd prime $p$, $\\gcd(s,p)=1$.\n\\end{itemize}\nThen, for the relevant factor $\\alpha(L) \\sqrt{M}$ listed above, the density of easy instances for our algorithm is at most $\\frac{\\alpha(L) \\log(M)}{\\sqrt{M}}$, which goes to zero as $M$ tends to infinity for all considered fields.\n\\\\ \\textbf{General Ideals and Modules:} For any of the distributions covered, we can make similar assertions about the density of easy cases in general ideals and modules. If the probability of choosing an easily solvable prime ideal lattice in a given distribution is $P$, then given a general ideal $\\mathcal{I}$ that has $g$ distinct factors, the probability that the ideal is easily solvable is $P^g$, since we require that all the prime ideal factors are individually easy cases. Similarly for the module case, for a module $M$ of rank $d$ over $\\mathcal{O}_L$, we have $M \\cong \\bigoplus_{i=1}^d\\mathcal{I}_i$. Then applying the previous logic over the prime decomposition of all the ideals $\\mathcal{I}_i$, if $\\mathcal{I}_i$ has $g_i$ distinct prime ideal factors, then the probability of picking an easily solvable module lattice is $P^{\\prod_{i=1}^dg_i}$.\n\\section{Concluding Remarks}\nIn this paper, we have successfully generalised methods pioneered by Pan et. al. in \\cite{idealrandomprime}. First, we showed that a solution to Hermite SVP for a general ideal lattice may be yielded by solving Hermite-SVP with a smaller factor in a subideal by exploiting the decomposition group of the ideal. Moreover, we showed that a similar argument may be made for module lattices defined over the ring of integers of a field that is Galois over $\\mathbb{Q}$, and present two methods by which we may approach the module case. For ideals of cyclotomic rings, we generalised Pan et. al.'s results to construct an efficient SVP algorithm for ideals of cyclotomic rings of arbitrary conductor, and showed that there exists certain classes of ideal lattices whose structure is significantly weaker to our algorithm than others. Moreover, we proved that our method may be generalised to the case of modules over arbitrary cyclotomic rings, which may have consequences for the security of MLWE. However, unlike the case for ideal lattices, the problem remains open to identify classes of modules over cyclotomic rings which may be weaker under attack by our algorithm.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}