An algorithmic approach using multivariate polynomials for the nonlinearity of Boolean functions

TL;DR

This paper introduces two practical algorithms for computing the nonlinearity of Boolean functions using multivariate polynomial systems, improving upon previous methods by enhancing efficiency and applicability.

Contribution

It presents novel, more efficient algorithms for solving polynomial systems related to Boolean function nonlinearity, extending the approach to different field characteristics.

Findings

Faster solution detection for polynomial systems in nonlinearity computation

Implementation of algorithms in MAGMA software

Enhanced applicability to various field characteristics

Abstract

The nonlinearity of a Boolean function is a key property in deciding its suitability for cryptographic purposes, e.g. as a combining function in stream ciphers, and so the nonlinearity computation is an important problem for applications. Traditional methods to compute the nonlinearity are based on transforms, such as the Fast Walsh Transform. In 2007 Simonetti proposed a method to solve the above problem seen as a decision problem on the existence of solutions for some multivariate polynomial systems. Although novel as approach, her algorithm suffered from a direct application of Groebner bases and was thus impractical. We now propose two more practical approaches, one that determines the existence of solutions for Simonetti's systems in a faster way and another that writes similar systems but over fields with a different characteristics. For our algorithms we provide an efficient implementation in the software package MAGMA.