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A class of weightwise almost perfectly balanced Boolean functions 一类权重几乎完全平衡的布尔函数
4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023048
Deepak Kumar Dalai, Krishna Mallick
{"title":"A class of weightwise almost perfectly balanced Boolean functions","authors":"Deepak Kumar Dalai, Krishna Mallick","doi":"10.3934/amc.2023048","DOIUrl":"https://doi.org/10.3934/amc.2023048","url":null,"abstract":"Constructing Boolean functions with good cryptographic properties over a subset of vectors with fixed Hamming weight $ E_{n,k} subset {{rm{I!F}}}_2^n $ is significant in lightweight stream ciphers like FLIP [14]. In this article, we have given a construction for a class of $ n $-variable weightwise almost perfectly balanced (WAPB) Boolean functions from known support of an $ n_0 $-variable WAPB Boolean function where $ n_0 < n $. This is a generalization of constructing a weightwise perfectly balanced (WPB) Boolean function by Mesnager and Su [16]. We have studied some cryptographic properties like ANF, nonlinearity, weightwise nonlinearities, and algebraic immunity of the functions. The ANF of this function is obtained recursively, which would be a low-cost implementation in a lightweight stream cipher. Further, we have presented another class of WAPB Boolean functions by modifying the earlier function, and we studied some of its cryptographic properties. The nonlinearity and weightwise nonlinearities of the modified functions improve substantially.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135709288","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}
引用次数: 0
Covering radius of $ RM(4,8) $ 覆盖半径$ RM(4,8) $
4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023038
Valérie Gillot, Philippe Langevin
{"title":"Covering radius of $ RM(4,8) $","authors":"Valérie Gillot, Philippe Langevin","doi":"10.3934/amc.2023038","DOIUrl":"https://doi.org/10.3934/amc.2023038","url":null,"abstract":"We propose an effective version of the lift by derivation, an invariant that allows us to provide the classification of $ B(5,6,8) = RM(6, 8)/RM(4, 8) $. The main consequence is to establish that the covering radius of the Reed-Muller $ RM(4,8) $ is equal to 26.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136258527","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}
引用次数: 0
Lee metrics on groups 李组度量
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-06-09 DOI: 10.3934/amc.2023011
Ricardo A. Podest'a, Maximiliano G. Vides
{"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":null,"pages":null},"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}
引用次数: 0
Introduction to the special issue dedicated to Cunsheng Ding on the occasion of his 60th birthday 丁存生60大寿特刊简介
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022089
{"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":null,"pages":null},"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}
引用次数: 0
Reconstructing points of superelliptic curves over a prime finite field 素数有限域上超椭圆曲线点的重构
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022022
J. Gutierrez
{"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":null,"pages":null},"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}
引用次数: 2
Optimal wide-gap-zone frequency hopping sequences 最优宽间隙区跳频序列
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022097
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":null,"pages":null},"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}
引用次数: 0
Three constructions of asymptotically optimal codebooks via multiplicative characters of finite fields 利用有限域的乘法特征构造三种渐近最优码本
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022091
{"title":"Three constructions of asymptotically optimal codebooks via multiplicative characters of finite fields","authors":"","doi":"10.3934/amc.2022091","DOIUrl":"https://doi.org/10.3934/amc.2022091","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81358670","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}
引用次数: 0
Finding small roots for bivariate polynomials over the ring of integers 寻找整数环上二元多项式的小根
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022012
Jiseung Kim, Changmin Lee
{"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":null,"pages":null},"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}
引用次数: 0
$ mathbb{Z}_{p^r}mathbb{Z}_{p^s}mathbb{Z}_{p^t} $-additive cyclic codes $ mathbb {Z} _ {p r ^} mathbb {Z} _ {p s ^} mathbb {Z} _ {p ^ t} $添加剂循环码
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022079
Raziyeh Molaei, K. Khashyarmanesh
{"title":"$ 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> $end{document}</tex-math></inline-formula>. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78895618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Constructions for several classes of few-weight linear codes and their applications 几类少权线性码的构造及其应用
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2022-01-01 DOI: 10.3934/amc.2022041
Canze Zhu, Qunying Liao
{"title":"Constructions for several classes of few-weight linear codes and their applications","authors":"Canze Zhu, Qunying Liao","doi":"10.3934/amc.2022041","DOIUrl":"https://doi.org/10.3934/amc.2022041","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, for any odd prime <inline-formula><tex-math id=\"M1\">begin{document}$ p $end{document}</tex-math></inline-formula> and an integer <inline-formula><tex-math id=\"M2\">begin{document}$ mge 3 $end{document}</tex-math></inline-formula>, several classes of linear codes with <inline-formula><tex-math id=\"M3\">begin{document}$ t $end{document}</tex-math></inline-formula>-weight <inline-formula><tex-math id=\"M4\">begin{document}$ (t = 3,5,7) $end{document}</tex-math></inline-formula> are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for <inline-formula><tex-math id=\"M5\">begin{document}$ m = 3 $end{document}</tex-math></inline-formula>, a class of MDS codes with parameters <inline-formula><tex-math id=\"M6\">begin{document}$ [p,3,p-2] $end{document}</tex-math></inline-formula> are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and <inline-formula><tex-math id=\"M7\">begin{document}$ s $end{document}</tex-math></inline-formula>-sum sets for any odd <inline-formula><tex-math id=\"M8\">begin{document}$ s>1 $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88310617","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}
引用次数: 3
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