On Cryptographic Parameters of Permutation Polynomials of the form x(r)h(x((2n-1)/d))

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(r)h(x((2n-1)/d)) over a finite field of q = 2(n) elements, where r is a positive integer and d is a positive divisor of 2(n-1). The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form x(r)h(x((2n-1)/3)) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6 <= n <= 12 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.