邻单项式和 2 对 1 二项式的分类是等价的

IF 1.2 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2024-07-30 DOI:10.1007/s10623-024-01463-1

Lukas Kölsch, Gohar Kyureghyan

{"title":"邻单项式和 2 对 1 二项式的分类是等价的","authors":"Lukas Kölsch, Gohar Kyureghyan","doi":"10.1007/s10623-024-01463-1","DOIUrl":null,"url":null,"abstract":"We observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials \$b=x^d+ux^e\$ for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"183 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The classifications of o-monomials and of 2-to-1 binomials are equivalent\",\"authors\":\"Lukas Kölsch, Gohar Kyureghyan\",\"doi\":\"10.1007/s10623-024-01463-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials \\\$b=x^d+ux^e\\\$ for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":\"183 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-07-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Designs, Codes and Cryptography\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01463-1\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01463-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}

引用次数: 0

摘要

我们观察到，在二元有限域上，2-to-1 二项式的分类等同于邻单项式的分类，而邻单项式的分类是有限几何中一个研究得很透彻而又难以捉摸的问题。这种联系意味着对于大量的 (d, e) 值集，2-to-1 二项式的完整分类是 \(b=x^d+ux^e/)。此外，我们还证明了一些已知的 2 对 1 映射无穷族可以追溯到邻多项式或 APN 映射的差映射。我们还提供了 2 到 1 映射与非德萨格平面中的超ovals 之间的一些联系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

The classifications of o-monomials and of 2-to-1 binomials are equivalent

We observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials $b=x^d+ux^e$ for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.

期刊最新文献

Line-parallelisms of PG(n, 2) from Preparata-like codes Evaluation codes from linear systems of conics $$\mathbb {F}_2\mathbb {F}_4$$-ACP of codes On codes from split metacyclic groups Hierarchical identity-based encryption with receiver selective opening security in the multi-challenge setting