Xiantong Tan | Searcharxiv

arXiv2022

Two low differentially uniform power permutations over odd characteristic finite fields: APN and differentially $4$-uniform functions

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.

cs.IT↗

arXiv2022

On the differential spectrum of a class of APN power functions over odd characteristic finite fields and their $c$-differential properties

Only three classes of Almost Perfect Nonlinear (for short, APN) power functions over odd characteristic finite fields have been investigated in the literature, and their differential spectra were determined. The differential uniformity of the power function $F(x)=x^{\frac{p^{n}-3}{2}}$ over the finite field $F_{p^n}$ of order $p^n$ (where $p$ is an odd prime), was studied by Helleseth and Sandberg in 1997, where $p^n\equiv3\pmod{4}$ is an odd prime power with $p^n>7$. It was shown that $F$ is PN when $p^n=27$, APN when $5$ is a nonsquare in $F_{p^n}$, and differentially $3$-uniform when $5$ is a square in $F_{p^n}$. In this paper, by investigating some equation systems and certain character sums over $F_{p^n}$, the differential spectrum of $F$ is completely determined. We focusing on the power functions $x^d$ with even $d$ over $F_{p^n}$ ($p$ odd), the power functions $F$ we consider are APN which are of the lowest differential uniformity and the nontrivial differential spectrum. Moreover, we examine the extension of the so-called $c$-differential uniformity by investigating the $c$-differential properties of $F$. Specifically, an upper bound of the $c$-differential uniformity of $F$ is given, and its $c$-differential spectrum is considered in the case where $c=-1$. Finally, we emphasize that, throughout our study of the differential spectrum of the considered power functions, we provide methods for evaluating sums of specific characters with connections to elliptic curves and for determining the number of solutions of specific systems of equations over finite fields.

cs.IT↗