We propose a decentralized e-voting protocol that is coercion-resistant and vote-selling resistant, while being also completely transparent and not receipt-free. We achieve decentralization using blockchain technology. Because of the properties such as transparency, decentralization, and non-repudiation, blockchain is a fundamental technology of great interest in its own right, and it also has large potential when integrated into many other areas. We prove the security of the protocol under the standard DDH assumption on the underlying prime-order cyclic group (e.g. the group of points of an elliptic curve), as well as under standard assumptions on blockchain robustness.
{"title":"A coercion-resistant blockchain-based E-voting protocol with receipts","authors":"Chiara Spadafora, Riccardo Longo, M. Sala","doi":"10.3934/AMC.2021005","DOIUrl":"https://doi.org/10.3934/AMC.2021005","url":null,"abstract":"We propose a decentralized e-voting protocol that is coercion-resistant and vote-selling resistant, while being also completely transparent and not receipt-free. We achieve decentralization using blockchain technology. Because of the properties such as transparency, decentralization, and non-repudiation, blockchain is a fundamental technology of great interest in its own right, and it also has large potential when integrated into many other areas. We prove the security of the protocol under the standard DDH assumption on the underlying prime-order cyclic group (e.g. the group of points of an elliptic curve), as well as under standard assumptions on blockchain robustness.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"80 1","pages":"500-521"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72713615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and to have not interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $G$ is a torsion-free group or a finite group of odd order, we prove that $G$ has a Lee metric if and only if $G$ is cyclic. Also, if $G$ is a group admitting Lee metrics then $G times mathbb{Z}_2^k$ always have Lee metrics for every $k in mathbb{N}$. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $le 31$ indicating which of them have (or have not) Lee metrics and why (not).
{"title":"Lee metrics on groups","authors":"Ricardo A. Podest'a, Maximiliano G. Vides","doi":"10.3934/amc.2023011","DOIUrl":"https://doi.org/10.3934/amc.2023011","url":null,"abstract":"In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and to have not interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $G$ is a torsion-free group or a finite group of odd order, we prove that $G$ has a Lee metric if and only if $G$ is cyclic. Also, if $G$ is a group admitting Lee metrics then $G times mathbb{Z}_2^k$ always have Lee metrics for every $k in mathbb{N}$. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $le 31$ indicating which of them have (or have not) Lee metrics and why (not).","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"94 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91299998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For applications in algebraic geometry codes, an explicit description of bases of the Riemann-Roch spaces over function fields is needed. We investigate the third function field begin{document}$ F^{(3)} $end{document} in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlăduţ bound. We construct new bases for the related Riemann-Roch spaces of begin{document}$ F^{(3)} $end{document} and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on begin{document}$ F^{(3)} $end{document}. All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).
For applications in algebraic geometry codes, an explicit description of bases of the Riemann-Roch spaces over function fields is needed. We investigate the third function field begin{document}$ F^{(3)} $end{document} in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlăduţ bound. We construct new bases for the related Riemann-Roch spaces of begin{document}$ F^{(3)} $end{document} and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on begin{document}$ F^{(3)} $end{document}. All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).
{"title":"Weierstrass semigroups on the third function field in a tower attaining the Drinfeld-Vlăduţ bound","authors":"Shudi Yang, Chuangqiang Hu","doi":"10.3934/amc.2022066","DOIUrl":"https://doi.org/10.3934/amc.2022066","url":null,"abstract":"<p style='text-indent:20px;'>For applications in algebraic geometry codes, an explicit description of bases of the Riemann-Roch spaces over function fields is needed. We investigate the third function field <inline-formula><tex-math id=\"M1\">begin{document}$ F^{(3)} $end{document}</tex-math></inline-formula> in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlăduţ bound. We construct new bases for the related Riemann-Roch spaces of <inline-formula><tex-math id=\"M2\">begin{document}$ F^{(3)} $end{document}</tex-math></inline-formula> and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on <inline-formula><tex-math id=\"M3\">begin{document}$ F^{(3)} $end{document}</tex-math></inline-formula>. All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"6 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87408615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Introduction to the special issue dedicated to Cunsheng Ding on the occasion of his 60th birthday","authors":"","doi":"10.3934/amc.2022089","DOIUrl":"https://doi.org/10.3934/amc.2022089","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"26 1","pages":"667-670"},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73241741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let begin{document}$ p $end{document} be a prime and begin{document}$ mathbb{F}_p $end{document} the finite field with begin{document}$ p $end{document} elements. We show how, when given an superelliptic curve begin{document}$ Y^n+f(X) in mathbb{F}_p[X,Y] $end{document} and an approximation to begin{document}$ (v_0,v_1) in mathbb{F}_p^2 $end{document} such that begin{document}$ v_1^n = -f(v_0) $end{document}, one can recover begin{document}$ (v_0,v_1) $end{document} efficiently, if the approximation is good enough. As consequence we provide an upper bound on the number of roots of such bivariate polynomials where the roots have certain restrictions. The results has been motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography.
Let begin{document}$ p $end{document} be a prime and begin{document}$ mathbb{F}_p $end{document} the finite field with begin{document}$ p $end{document} elements. We show how, when given an superelliptic curve begin{document}$ Y^n+f(X) in mathbb{F}_p[X,Y] $end{document} and an approximation to begin{document}$ (v_0,v_1) in mathbb{F}_p^2 $end{document} such that begin{document}$ v_1^n = -f(v_0) $end{document}, one can recover begin{document}$ (v_0,v_1) $end{document} efficiently, if the approximation is good enough. As consequence we provide an upper bound on the number of roots of such bivariate polynomials where the roots have certain restrictions. The results has been motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography.
{"title":"Reconstructing points of superelliptic curves over a prime finite field","authors":"J. Gutierrez","doi":"10.3934/amc.2022022","DOIUrl":"https://doi.org/10.3934/amc.2022022","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ p $end{document}</tex-math></inline-formula> be a prime and <inline-formula><tex-math id=\"M2\">begin{document}$ mathbb{F}_p $end{document}</tex-math></inline-formula> the finite field with <inline-formula><tex-math id=\"M3\">begin{document}$ p $end{document}</tex-math></inline-formula> elements. We show how, when given an superelliptic curve <inline-formula><tex-math id=\"M4\">begin{document}$ Y^n+f(X) in mathbb{F}_p[X,Y] $end{document}</tex-math></inline-formula> and an approximation to <inline-formula><tex-math id=\"M5\">begin{document}$ (v_0,v_1) in mathbb{F}_p^2 $end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id=\"M6\">begin{document}$ v_1^n = -f(v_0) $end{document}</tex-math></inline-formula>, one can recover <inline-formula><tex-math id=\"M7\">begin{document}$ (v_0,v_1) $end{document}</tex-math></inline-formula> efficiently, if the approximation is good enough. As consequence we provide an upper bound on the number of roots of such bivariate polynomials where the roots have certain restrictions. The results has been motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"55 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72662037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let begin{document}$ p $end{document} be a prime number and begin{document}$ r, s, t $end{document} be positive integers such that begin{document}$ rle sle t $end{document}. A begin{document}$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $end{document}-additive code is a begin{document}$ mathbb{Z}_{p^t} $end{document}-submodule of begin{document}$ mathbb{Z}_{p^r}^{alpha} times mathbb{Z}_{p^s}^{beta} times mathbb{Z}_{p^t}^{gamma} $end{document}, where begin{document}$ alpha, beta, gamma $end{document} are positive integers. In this paper, we study begin{document}$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $end{document}-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring begin{document}$ R = mathbb{Z}_{p^r}[x]/big times mathbb{Z}_{p^s}[x]/big times mathbb{Z}_{p^t}[x]/big $end{document}. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.
Let begin{document}$ p $end{document} be a prime number and begin{document}$ r, s, t $end{document} be positive integers such that begin{document}$ rle sle t $end{document}. A begin{document}$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $end{document}-additive code is a begin{document}$ mathbb{Z}_{p^t} $end{document}-submodule of begin{document}$ mathbb{Z}_{p^r}^{alpha} times mathbb{Z}_{p^s}^{beta} times mathbb{Z}_{p^t}^{gamma} $end{document}, where begin{document}$ alpha, beta, gamma $end{document} are positive integers. In this paper, we study begin{document}$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $end{document}-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring begin{document}$ R = mathbb{Z}_{p^r}[x]/big times mathbb{Z}_{p^s}[x]/big times mathbb{Z}_{p^t}[x]/big $end{document}. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.
{"title":"$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $-additive cyclic codes","authors":"Raziyeh Molaei, K. Khashyarmanesh","doi":"10.3934/amc.2022079","DOIUrl":"https://doi.org/10.3934/amc.2022079","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M3\">begin{document}$ p $end{document}</tex-math></inline-formula> be a prime number and <inline-formula><tex-math id=\"M4\">begin{document}$ r, s, t $end{document}</tex-math></inline-formula> be positive integers such that <inline-formula><tex-math id=\"M5\">begin{document}$ rle sle t $end{document}</tex-math></inline-formula>. A <inline-formula><tex-math id=\"M6\">begin{document}$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $end{document}</tex-math></inline-formula>-additive code is a <inline-formula><tex-math id=\"M7\">begin{document}$ mathbb{Z}_{p^t} $end{document}</tex-math></inline-formula>-submodule of <inline-formula><tex-math id=\"M8\">begin{document}$ mathbb{Z}_{p^r}^{alpha} times mathbb{Z}_{p^s}^{beta} times mathbb{Z}_{p^t}^{gamma} $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M9\">begin{document}$ alpha, beta, gamma $end{document}</tex-math></inline-formula> are positive integers. In this paper, we study <inline-formula><tex-math id=\"M10\">begin{document}$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $end{document}</tex-math></inline-formula>-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring <inline-formula><tex-math id=\"M11\">begin{document}$ R = mathbb{Z}_{p^r}[x]/big<x^alpha-1big> times mathbb{Z}_{p^s}[x]/big<x^beta-1big> times mathbb{Z}_{p^t}[x]/big<x^gamma-1big> $end{document}</tex-math></inline-formula>. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"11 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78895618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Qin Shu, Hai Liu, Xing Liu, Yun-xiu Yang, Wendong Chen
{"title":"Optimal wide-gap-zone frequency hopping sequences","authors":"Qin Shu, Hai Liu, Xing Liu, Yun-xiu Yang, Wendong Chen","doi":"10.3934/amc.2022097","DOIUrl":"https://doi.org/10.3934/amc.2022097","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"25 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81436872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct self-dual codes over begin{document}$ mathbb{F}_{11} $end{document}, begin{document}$ mathbb{F}_{13} $end{document}, begin{document}$ mathbb{F}_{17} $end{document}, begin{document}$ mathbb{F}_{19} $end{document} and begin{document}$ mathbb{F}_{23} $end{document} which improve the previously known lower bounds on the largest minimum weights. In particular, the largest possible minimum weight among self-dual begin{document}$ [n, n/2] $end{document} codes over begin{document}$ mathbb{F}_{p} $end{document} is determined for begin{document}$ (p, n) = (19, 24) $end{document} and begin{document}$ (23, 28) $end{document}.
We construct self-dual codes over begin{document}$ mathbb{F}_{11} $end{document}, begin{document}$ mathbb{F}_{13} $end{document}, begin{document}$ mathbb{F}_{17} $end{document}, begin{document}$ mathbb{F}_{19} $end{document} and begin{document}$ mathbb{F}_{23} $end{document} which improve the previously known lower bounds on the largest minimum weights. In particular, the largest possible minimum weight among self-dual begin{document}$ [n, n/2] $end{document} codes over begin{document}$ mathbb{F}_{p} $end{document} is determined for begin{document}$ (p, n) = (19, 24) $end{document} and begin{document}$ (23, 28) $end{document}.
{"title":"Improved lower bounds for self-dual codes over $ mathbb{F}_{11} $, $ mathbb{F}_{13} $, $ mathbb{F}_{17} $, $ mathbb{F}_{19} $ and $ mathbb{F}_{23} $","authors":"T. Gulliver, M. Harada","doi":"10.3934/amc.2022083","DOIUrl":"https://doi.org/10.3934/amc.2022083","url":null,"abstract":"<p style='text-indent:20px;'>We construct self-dual codes over <inline-formula><tex-math id=\"M6\">begin{document}$ mathbb{F}_{11} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M7\">begin{document}$ mathbb{F}_{13} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M8\">begin{document}$ mathbb{F}_{17} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M9\">begin{document}$ mathbb{F}_{19} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M10\">begin{document}$ mathbb{F}_{23} $end{document}</tex-math></inline-formula> which improve the previously known lower bounds on the largest minimum weights. In particular, the largest possible minimum weight among self-dual <inline-formula><tex-math id=\"M11\">begin{document}$ [n, n/2] $end{document}</tex-math></inline-formula> codes over <inline-formula><tex-math id=\"M12\">begin{document}$ mathbb{F}_{p} $end{document}</tex-math></inline-formula> is determined for <inline-formula><tex-math id=\"M13\">begin{document}$ (p, n) = (19, 24) $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M14\">begin{document}$ (23, 28) $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"25 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83277931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose the first heuristic algorithm for finding small roots for a bivariate equation modulo an ideal begin{document}$ mathcal{I} $end{document} over the ring of integers begin{document}$ mathcal{R} $end{document}. Existing algorithms for solving polynomial equations with size constraints only work for bivariate modular equations over integers, and univariate modular equation over number fields.
Both previous algorithms use a relation between the short vector in a skillfully structured lattice and a size constrained solution. Our algorithm also follows this framework, but we additionally use a polynomial factoring algorithm over number fields to recover a 'ring' root of a bivariate polynomial equation.
As a result, when an LLL algorithm is employed to find a short vector, we can recover all small roots of a bivariate polynomial modulo begin{document}$ mathcal{I} $end{document} in polynomial time under some constraint.
In this paper, we propose the first heuristic algorithm for finding small roots for a bivariate equation modulo an ideal begin{document}$ mathcal{I} $end{document} over the ring of integers begin{document}$ mathcal{R} $end{document}. Existing algorithms for solving polynomial equations with size constraints only work for bivariate modular equations over integers, and univariate modular equation over number fields.Both previous algorithms use a relation between the short vector in a skillfully structured lattice and a size constrained solution. Our algorithm also follows this framework, but we additionally use a polynomial factoring algorithm over number fields to recover a 'ring' root of a bivariate polynomial equation.As a result, when an LLL algorithm is employed to find a short vector, we can recover all small roots of a bivariate polynomial modulo begin{document}$ mathcal{I} $end{document} in polynomial time under some constraint.
{"title":"Finding small roots for bivariate polynomials over the ring of integers","authors":"Jiseung Kim, Changmin Lee","doi":"10.3934/amc.2022012","DOIUrl":"https://doi.org/10.3934/amc.2022012","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we propose the first heuristic algorithm for finding small roots for a bivariate equation modulo an ideal <inline-formula><tex-math id=\"M3\">begin{document}$ mathcal{I} $end{document}</tex-math></inline-formula> over the ring of integers <inline-formula><tex-math id=\"M4\">begin{document}$ mathcal{R} $end{document}</tex-math></inline-formula>. Existing algorithms for solving polynomial equations with size constraints only work for bivariate modular equations over integers, and univariate modular equation over number fields.</p><p style='text-indent:20px;'>Both previous algorithms use a relation between the short vector in a skillfully structured lattice and a size constrained solution. Our algorithm also follows this framework, but we additionally use a polynomial factoring algorithm over number fields to recover a 'ring' root of a bivariate polynomial equation.</p><p style='text-indent:20px;'>As a result, when an LLL algorithm is employed to find a short vector, we can recover all small roots of a bivariate polynomial modulo <inline-formula><tex-math id=\"M5\">begin{document}$ mathcal{I} $end{document}</tex-math></inline-formula> in polynomial time under some constraint.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"23 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82463560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：