{"title":"Differential uniformity properties of some classes of permutation polynomials","authors":"Kirpa Garg , Sartaj Ul Hasan , Pantelimon Stănică","doi":"10.1016/j.jaca.2025.100035","DOIUrl":null,"url":null,"abstract":"<div><div>The notion of <em>c</em>-differential uniformity has received a lot of attention since its proposal <span><span>[5]</span></span>, and recently a characterization of perfect <em>c</em>-nonlinear functions in terms of difference sets in some quasigroups was obtained in <span><span>[1]</span></span>. Moreover, in a very recent manuscript by Pal and Stănică <span><span>[19]</span></span>, an intriguing connection was discovered showing that in fact, the boomerang uniformity for an odd APN function (odd characteristic) equals its <em>c</em>-differential uniformity when <span><math><mi>c</mi><mo>=</mo><mo>−</mo><mn>1</mn></math></span>, if the function is a permutation, otherwise it is the maximum of the <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-DDT entries disregarding the first row/column. The construction of functions, especially permutations, with low <em>c</em>-differential uniformity is an interesting and difficult mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low <em>c</em>-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest, from a mathematical perspective.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"13 ","pages":"Article 100035"},"PeriodicalIF":0.0000,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational Algebra","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2772827725000063","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The notion of c-differential uniformity has received a lot of attention since its proposal [5], and recently a characterization of perfect c-nonlinear functions in terms of difference sets in some quasigroups was obtained in [1]. Moreover, in a very recent manuscript by Pal and Stănică [19], an intriguing connection was discovered showing that in fact, the boomerang uniformity for an odd APN function (odd characteristic) equals its c-differential uniformity when , if the function is a permutation, otherwise it is the maximum of the -DDT entries disregarding the first row/column. The construction of functions, especially permutations, with low c-differential uniformity is an interesting and difficult mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low c-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest, from a mathematical perspective.