Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

TL;DR

This paper presents a new technique for constructing highly nonlinear, almost optimal resilient Boolean functions on large even-variable sets, expanding the class of known functions with strong cryptographic properties.

Contribution

It introduces a method to construct infinitely many resilient Boolean functions with high nonlinearity on large even numbers of variables, including an optimization approach and an improved construction.

Findings

Constructed resilient functions with nonlinearity > 2^{n-1}-2^{n/2}

Generated a large class of previously unknown highly nonlinear resilient functions

Proposed an optimization method for the degree of the functions

Abstract

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.