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

IF 1.2 3区 数学 Q1 MATHEMATICS
Daniel Smith-Tone , Cristina Tone
{"title":"A correct justification for the CHMT algorithm for solving underdetermined multivariate systems","authors":"Daniel Smith-Tone ,&nbsp;Cristina Tone","doi":"10.1016/j.ffa.2024.102547","DOIUrl":null,"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":1.2000,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724001862","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Cheng et al. (2014) [6] 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 n(m+12), where m is the number of equations and n 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.
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.
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
2.00
自引率
20.00%
发文量
133
审稿时长
6-12 weeks
期刊介绍: Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信