Construction of 1-Resilient Boolean Functions with Very Good Nonlinearity

Abstract

In this paper we present a strategy to construct 1-resilient Boolean functions with very good nonlinearity and autocorrelation. Our strategy to construct a 1-resilient function is based on modifying a bent function, by toggling some of its output bits. Two natural questions that arise in this context are “at least how many bits and which bits in the output of a bent function need to be changed to construct a 1-resilient Boolean function”. We present an algorithm which determines a minimum number of bits of a bent function that need to be changed to construct a 1-resilient Boolean function. We also present a technique to compute points whose output in the bent function need to be modified to get a 1-resilient function. In particular, the technique is applied upto 14-variable functions and we show that the construction provides 1-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation absolute indicator values which were not known earlier.