Finite Fields and Their Applications最新文献_第4页

A correct justification for the CHMT algorithm for solving underdetermined multivariate systems

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-12-11 DOI: 10.1016/j.ffa.2024.102547

Daniel Smith-Tone , Cristina Tone

{"title":"A correct justification for the CHMT algorithm for solving underdetermined multivariate systems","authors":"Daniel Smith-Tone , Cristina Tone","doi":"10.1016/j.ffa.2024.102547","DOIUrl":"10.1016/j.ffa.2024.102547","url":null,"abstract":"<div><div>Cheng et al. (2014) <span><span>[6]</span></span> introduced a substantial improvement to the Miura-Hashimoto-Takagi algorithm for solving sufficiently underdetermined systems of multivariate polynomial equations. This improvement claimed to make the algorithm polynomial time for instances satisfying <span><math><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, where <em>m</em> is the number of equations and <em>n</em> is the number of variables. While experimentally, the algorithm seems to work, we have uncovered a subtle error in the proof of time complexity for the algorithm. Due to the fact that there have been multiple proposals for algorithms based on this and related algorithms, as well as the recent submission to NIST's call for additional post-quantum digital signatures of a more modern “provably secure” version of the famous UOV digital signature algorithm based on the foundational structure of this algorithm, our observation may highlight a concerning theoretical deficiency in this area of research.</div><div>In this work, we provide a tight justification for the polynomial time complexity of the algorithm (with a very minor tweak), thereby justifying the complexity of enhancements based upon it as well. At the heart of this justification is a precise calculation of the probability of recovering a maximal depth path in polynomially many steps within a possibly exponentially large search tree. While this algorithmic problem is generic, we find that the parameters relevant for the application to the above algorithm are extremal and poorly studied. Thus, our analysis serves to clarify the boundary behavior of such search algorithms with respect to complexity classes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102547"},"PeriodicalIF":1.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the square code of group codes

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-12-10 DOI: 10.1016/j.ffa.2024.102548

Alejandro Piñera Nicolás , Ignacio Fernández Rúa , Adriana Suárez Corona

引用次数: 0

The R-transform as power map and its generalizations to higher degree 作为幂图的 R 变换及其对更高程度的概括

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-27 DOI: 10.1016/j.ffa.2024.102546

Alp Bassa , Ricardo Menares

{"title":"The R-transform as power map and its generalizations to higher degree","authors":"Alp Bassa , Ricardo Menares","doi":"10.1016/j.ffa.2024.102546","DOIUrl":"10.1016/j.ffa.2024.102546","url":null,"abstract":"<div><div>We give iterative constructions for irreducible polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi><mo>⋅</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for all <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, starting from irreducible polynomials of degree <em>n</em>. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span>. The <em>R</em>-transform introduced by Cohen is recovered as a particular case corresponding to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, hence we obtain a generalization of Cohen's <em>R</em>-transform (<span><math><mi>t</mi><mo>=</mo><mn>2</mn></math></span>) to arbitrary degrees <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> we recover and generalize a recursive construction of Panario, Reis and Wang.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102546"},"PeriodicalIF":1.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Webs and squabs of conics over finite fields 有限域上圆锥的网和方锥

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-26 DOI: 10.1016/j.ffa.2024.102544

Nour Alnajjarine , Michel Lavrauw

{"title":"Webs and squabs of conics over finite fields","authors":"Nour Alnajjarine , Michel Lavrauw","doi":"10.1016/j.ffa.2024.102544","DOIUrl":"10.1016/j.ffa.2024.102544","url":null,"abstract":"<div><div>This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, for <em>q</em> odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, both for <em>q</em> odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The main contribution is the determination of the distribution of the different types of hyperplanes incident with the <em>K</em>-orbit representatives of points and lines of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>K</mi><mo>≅</mo><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, is the subgroup of <span><math><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> stabilizing the Veronese surface.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102544"},"PeriodicalIF":1.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Some q-ary constacyclic BCH codes with length qm+12 长度为 qm+12 的一些 qary 常环 BCH 码

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-21 DOI: 10.1016/j.ffa.2024.102545

Jin Li , Huilian Zhu , Shixin Zhu

引用次数: 0

Repeated-root constacyclic codes of length kslmpn over finite fields 有限域上长度为 kslmpn 的重复根常环码

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-14 DOI: 10.1016/j.ffa.2024.102542

Qi Zhang , Weiqiong Wang , Shuyu Luo , Yue Li

引用次数: 0

Complete description of measures corresponding to Abelian varieties over finite fields 有限域上阿贝尔变种对应度量的完整描述

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-14 DOI: 10.1016/j.ffa.2024.102543

Nikolai S. Nadirashvili , Michael A. Tsfasman

引用次数: 0

Self-dual 2-quasi negacyclic codes over finite fields 有限域上的自偶 2-quasi 负环码

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-07 DOI: 10.1016/j.ffa.2024.102541

Yun Fan, Yue Leng

引用次数: 0

Intersecting families of polynomials over finite fields 有限域上多项式的相交族

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-07 DOI: 10.1016/j.ffa.2024.102540

Nika Salia , Dávid Tóth

{"title":"Intersecting families of polynomials over finite fields","authors":"Nika Salia , Dávid Tóth","doi":"10.1016/j.ffa.2024.102540","DOIUrl":"10.1016/j.ffa.2024.102540","url":null,"abstract":"<div><div>This paper demonstrates an analog of the Erdős–Ko–Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins.</div><div>A <em>k</em>-uniform family of subsets of a set of size <em>n</em> is <em>ℓ</em>-intersecting if any two subsets in the family intersect in at least <em>ℓ</em> elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erdős–Ko–Rado theorem, which establishes a sharp upper bound on the size of these families. Here, we extend the Erdős–Ko–Rado theorem to polynomial rings over finite fields.</div><div>Specifically, we determine the largest possible size of a family of monic polynomials, each of degree <em>n</em>, over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where every pair of polynomials in the family shares a common factor of degree at least <em>ℓ</em>. We prove that the upper bound for this size is <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>ℓ</mi></mrow></msup></math></span> and characterize all extremal families that achieve this maximum size.</div><div>Extending to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least <em>ℓ</em>, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most <em>n</em>, we affirm that only trivial families achieve the corresponding upper bound.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102540"},"PeriodicalIF":1.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Partial difference sets with Denniston parameters in elementary abelian p-groups 初等无性 p 群中具有丹尼斯顿参数的部分差集

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2024-11-07 DOI: 10.1016/j.ffa.2024.102539

Jingjun Bao , Qing Xiang , Meng Zhao

引用次数: 0