Differential and boomerang spectrums of some power permutations

Abstract

The differential (resp. boomerang) spectrum is an important parameter to estimate the resistance of cryptographic functions against some variants of differential (resp. boomerang) cryptanalysis. This paper aims to determine the differential and boomerang spectrums of some power permutations. In 1997, Helleseth and Sandberg proved that the differential uniformity of $x^{\\frac {p^{n}-1}{2}+2}$\n over $\\mathbb {F}_{p^{n}}$\n , where p is an odd prime, is less than or equal to 4. In this paper, we first determine the differential spectrum of $x^{\\frac {3^{n}-1}{2}+2}$\n over $\\mathbb {F}_{3^{n}}$\n with n odd and then compute its boomerang spectrum based on the differential spectrum. In addition, in 2018, Boura and Canteaut determined the boomerang spectrum of the inverse function over $\\mathbb {F}_{2^{n}}$\n with n even. Following their work, we characterize the boomerang spectrum of the inverse function $x^{p^{n}-2}$\n over $\\mathbb {F}_{p^{n}}$\n for any odd prime p.