The differential properties of certain permutation polynomials over finite fields

2023-10-31

arXiv.org

AI Summary

This work addresses the construction of finite field permutation polynomials with low *c*-differential uniformity, motivated by the need for robust resistance against differential cryptanalysis in cryptography and combinatorial design. Focusing on finite fields of even characteristic and characteristic three, we propose four new families of perfect *c*-nonlinear (PCN) permutations and, for the first time, identify a class of permutations over characteristic-three fields with *c*-differential uniformity exactly equal to 3. Our approach integrates number-theoretic analysis, explicit computation of Walsh transform coefficients, and Weil sum estimation techniques to precisely solve the underlying functional equations. The results substantially expand the extremely scarce pool of known PCN functions; the constructed polynomials achieve optimal or near-optimal *c*-differential uniformity, and several constructions possess intrinsic theoretical significance. Collectively, this work provides essential building blocks and analytical tools for *c*-differential cryptanalysis.

Abstract

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown by Anbar et al. (J. Comb. Des. 31(12):1-24, 2023). Additionally, in a recent manuscript by Pal et al. (Adv. Math. Communications, to appear), a very interesting connection between the $c$-differential uniformity and boomerang uniformity, when $c=-1$, was pointed out, showing that they are the same for an odd APN permutation, sparking yet more interest in the construction of functions with low $c$-differential uniformity. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. To solve the involved equations over finite fields, we use various number theoretical techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

Problem

• Low c-differential uniformity
• Permutation polynomials
• Finite fields

Innovation

• Permutation Polynomials
• Low c-Differential Uniformity
• Walsh Transform Coefficients