2022/10/18 by Haode Yan, Sihem Mesnager, Yan, Haode +3
Computer Science · Social Sciences · #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Computer and information sciences #Information Theory (cs.IT) #Islamic Finance and Communication
paper · pdf · doi:10.48550/arxiv.2210.09822
openalex publication_date 2022/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(pn+3)/(2) defined over the finite field Fpn of order pn, 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 \citeHS by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of F when pn≡3\pmod 4 and p≠ 3. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some exponential sums over Fpn (which amounts to evaluating the number of Fpn-rational points on some related curves and secondly by computing the number of solutions in (Fpn)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 (pn≡3\pmod 4 and p≠ 3), F is an APN power permutation when pn=11, and a differentially 4-uniform power permutation otherwise.