论文标题
两个低差异均匀的功率排列在奇数特征有限场上：APN和差异$ 4 $均匀的功能
Two low differentially uniform power permutations over odd characteristic finite fields: APN and differentially $4$-uniform functions
论文作者
论文摘要
有限领域的置换多项式是基本对象，因为它们在密码学，编码理论，组合设计和相关主题的各种理论和实际应用中使用。这个多项式家族构成了一个积极的研究领域，在该领域中不断提高。特别是，尽管多年来在这个方向上进行了大量研究，但在具有良好差分特性的有限磁场上构建无限类别的多项式多项式（即低）仍然是一个令人兴奋的问题。 本文在有限特征的有限磁场上表现出低差异均匀的功率排列。具体而言，它的目标是双重的，关于功能函数$ f（x）= x^{\ frac {p^n+3} {2}} $在有限的字段$ f_ {p^n} $上定义的订单$ p^n $，其中$ p $，$ n $是奇怪的prime，$ n $是一个正面的integer。首先是补充Helleseth和Sandberg在\ cite {HS}中发起的一些以前的结果，通过解决开放式问题的开放时间超过二十年，即确定$ f $的差异频谱时，当$ p^n \ equiv3 \ equiv3 \ pmod 4 $和$ p \ p \ p \ neq 3 $。第二个是确定其差异均匀性的确切值。 Our achievements are obtained firstly by evaluating some exponential sums over $F_{p^n}$ (which amounts to evaluating the number of $F_{p^n}$-rational points on some related curves and secondly by computing the number of solutions in $(F_{p^n})^4$ of a system of equations presented by Helleseth, Rong, and Sandberg in ["New families of almost perfect nonlinear power mappings," IEEE Trans. Inform. Theory, vol. 45. no. 2, 1999], naturally appears while determining the differential spectrum of $F$. We show that in the considered case ($p^n\equiv3\pmod 4$ and $p\neq 3$), $F$ is an APN power permutation when $p^n=11$, and a differentially $4$-uniform power permutation otherwise.
Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristic. Specifically, its objective is twofold concerning the power functions $F(x)=x^{\frac{p^n+3}{2}}$ defined over the finite field $F_{p^n}$ of order $p^n$, where $p$ is an odd prime, and $n$ is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in \cite{HS} by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of $F$ when $p^n\equiv3\pmod 4$ and $p\neq 3$. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some exponential sums over $F_{p^n}$ (which amounts to evaluating the number of $F_{p^n}$-rational points on some related curves and secondly by computing the number of solutions in $(F_{p^n})^4$ of a system of equations presented by Helleseth, Rong, and Sandberg in ["New families of almost perfect nonlinear power mappings," IEEE Trans. Inform. Theory, vol. 45. no. 2, 1999], naturally appears while determining the differential spectrum of $F$. We show that in the considered case ($p^n\equiv3\pmod 4$ and $p\neq 3$), $F$ is an APN power permutation when $p^n=11$, and a differentially $4$-uniform power permutation otherwise.
