Nonlinear vector resilient functions

Abstract

An (n, m, k)-resilient function is a function f: F2n → F2m such that every possible output m-tuple is equally likely to occur when the values of k arbitrary inputs are fixed by an adversary and the remaining n - k input bits are chosen independently at random. In this paper we propose a new method to generate a (n + D + 1,m,d - 1)-resilient function for any non-negative integer D whenever a [n,m,d] linear code exists. This function has algebraic degree D and nonlinearity at least 2n+D - 2n⌊√2n+D+1r+ 2 n-1. If we apply this method to the simplex code, we can get a (t(2m - 1) + D + 1,m,t2m-1 - 1)-resilient function with algebraic degree D for any positive integers m,t and D. Note that if we increase the input size by D in the proposed construction, we can get a resilient function with the same parameter except algebraic degree increased by D. © Springer-Verlag Berlin Heidelberg 2001.