Differential uniformity properties of some classes of permutation polynomials

Kirpa Garg , Sartaj Ul Hasan , Pantelimon Stănică
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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 c=1, if the function is a permutation, otherwise it is the maximum of the (1)-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.
几类置换多项式的微分均匀性
自提出[5]以来,c-微分均匀性的概念受到了广泛的关注,最近在[1]中得到了一些准群中完备c-非线性函数在差分集上的刻画。此外,在Pal和striturnicei[19]最近的一篇手稿中,发现了一个有趣的联系,表明事实上,当c= - 1时,奇数APN函数(奇数特征)的回旋均匀性等于它的c-微分均匀性,如果函数是一个排列,否则它是(- 1)-DDT项的最大值,不管第一行/列。低c微分均匀性函数的构造,特别是排列的构造,是该领域中一个有趣而困难的数学问题,最近的工作主要集中在这个方向上。我们提供了几类低c微分均匀性的置换多项式。所使用的技术包括处理各种Weil和,以及分析有限域中的一些方程,我们相信从数学的角度来看,这些可能是独立的兴趣。
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