Yet another algorithm to compute the nonlinearity of a Boolean function

TL;DR

This paper introduces a polynomial-based method to compute the nonlinearity of Boolean functions, matching the complexity of existing algorithms like the fast Walsh transform, offering a new perspective on an important cryptographic measure.

Contribution

It presents a novel polynomial approach to determine Boolean function nonlinearity, aligning with the complexity of established methods.

Findings

Polynomial evaluation requires $O(n2^n)$ operations

Method matches the complexity of fast Walsh transform-based algorithms

Provides an alternative perspective on nonlinearity computation

Abstract

We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function and computing its evaluation vector requires a worst-case complexity of $O(n2^n)$ integer operations. This way, with a different approach, we reach the same complexity of established algorithms, such as those based on the fast Walsh transform.