An improvement of correlation analysis for vectorial boolean functions

Abstract

This paper investigates the correlation of n-bit to m-bit vectorial Boolean functions denoted by F. At Crypto 2000, Zhang and Chan showed that the maximum of linear approximations for F with Boolean functions g have a higher bias than those based on the usual correlation attack. The correlation for this linear approximation has been named the maximum correlation and has been shown to be a useful tool for correlation attack resistance. In this work, we deal with two issues. Firstly, we show that combining F with any g does not always increase the bias as stated by several works. To justify such results, we demonstrate the exact correlation link between F, g and the combination of F by g. Secondly, we provide the exact condition in which the correlation coefficients for this approximation are maximum.