On the max min of the algebraic degree and the nonlinearity of a Boolean function on an affine subspace

TL;DR

This paper studies the extremal behavior of algebraic degree and nonlinearity of Boolean functions when restricted to affine subspaces, providing bounds and conjectures for their maximum and minimum values.

Contribution

It introduces bounds and conjectures for the max min of algebraic degree and nonlinearity of Boolean functions on affine subspaces, extending previous work.

Findings

Upper and lower bounds for max min algebraic degree and nonlinearity.

Conjectures on the exact values in specific cases.

Focus on cases where max min algebraic degree is 0 or 1.

Abstract

We investigate the max min of the algebraic degree and the nonlinearity of a Boolean function in $n$ variables when restricted to a $k$-dimensional affine subspace of $\mathbb{F}_2^n$. Previous authors have focused on the cases when the max min of the algebraic degree is 0 or 1. Upper bounds, lower bounds and a conjecture on the exact value in special cases are presented.