Two results on maximum nonlinear functions

Abstract

Maximum nonlinear functions $F: \mathbb F_{2^m}\to \mathbb F_{2^m}$ are widely used in cryptography because the coordinate functions F _β (x) := tr(β F(x)), $\beta \in \mathbb F^{*}_{2^m}$ , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, $a,b \in \mathbb F_{2^m}, b\neq 0$ , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions $x^{2^{k}+1}$ , gcd(k, m) = 1, and the Kasami power functions $x^{2^{2k}-2^{k}+1}$ , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in $\mathbb F_{2^m}$ . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.