On a new notion of nonlinearity relevant to multi-output pseudo-random generators

Abstract

Vectorial functions (i.e. mappings from double struck F sign2n into double struck F sign2m, also called S-boxes) can be used in pseudo-random generators with multiple outputs. The notion of maximum correlation of these S-boxes to linear functions, introduced by Zhang and Chan, plays a central role in the resistance of the resulting stream ciphers to correlation attacks. It can be related to a notion of "unrestricted nonlinearity". We obtain a new lower bound on the overall maximum correlation to linear functions of vectorial functions which results in an upper bound on the unrestricted nonlinearity. We compare it with the known upper bounds on the nonlinearity (which are also valid for the unrestricted nonlinearity of balanced functions). We study its tightness and we exhibit a class of balanced functions whose nonlinearity and unrestricted nonlinearity are high relatively to the upper-bounds. © Springer-Verlag Berlin Heidelberg 2004.