Nonlinarity of Boolean functions and hyperelliptic curves

TL;DR

This paper investigates the nonlinearity of Boolean functions derived from polynomials over finite fields, revealing that functions with odd field degrees exhibit strong nonlinearity properties, supported by algebraic geometry results.

Contribution

It connects the nonlinearity of certain Boolean functions to properties of supersingular curves, providing new criteria for almost perfect nonlinear functions.

Findings

Functions with odd m have good nonlinearity properties.

Uses results on zeta functions of supersingular curves.

Provides criteria for non-APN vectorial functions.

Abstract

We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent results of Maisner and Nart about zeta functions of supersingular curves of genus 2. We give some criterion for a vectorial function not to be almost perfect nonlinear.