Spectral and Nonlinear Properties of the Sum of Boolean Functions

Abstract

Boolean functions are the mathematical basis of modern cryptographic algorithms. However, in practice, a set of interrelated Boolean functions is often used to construct a cryptographic algorithm. This circumstance makes the task of research of cryptographic quality, in particular, the distance of the nonlinearity of the sum of few Boolean functions important. The nonlinearity distance of a Boolean function is determined by the maximum value of its Walsh-Hadamard transform coefficients. In this paper, we proposed a formula that is the equivalent of the summation of Boolean functions in the Walsh-Hadamard transform domain. The application of this formula, as well as the Walsh-Hadamard spectral classification made it possible to determine the structure of WalshHadamard transform coefficients, and the distance of the nonlinearity when summing the Boolean functions lengths N = 8 and N = 16, indicating valuable practical application for information protection.