Paper 2019/767

On cryptographic parameters of permutation polynomials of the form $x^rh(x^{(q-1)/d})$

Jaeseong Jeong, Chang Heon Kim, Namhun Koo, Soonhak Kwon, and Sumin Lee

Abstract

The differential uniformity, the boomerang uniformity, and the extended Walsh spectrum etc are important parameters to evaluate the security of S(substitution)-box. In this paper, we introduce efficient formulas to compute these cryptographic parameters of permutation polynomials of the form $x^rh(x^{(q-1)/d})$ over a finite field of $q=2^n$ elements, where $r$ is a positive integer and $d$ is a positive divisor of $q-1$. The computational cost of those formulas is proportional to $d$. We investigate differentially 4-uniform permutation polynomials of the form $x^rh(x^{(q-1)/3})$ and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when $4\le n\le 10$ is even, where $d=3$ is the smallest nontrivial $d$ for even $n$. We also investigate the differential uniformity of some permutation polynomials introduced in some recent papers for the case $d=2^{n/2}+1$