On the max min of the algebraic degree and the nonlinearity of a Boolean function on an affine subspace
Jan Kristian Haugland
TL;DR
This work studies how large the minimal algebraic degree $\\alpha(f,k)$ and the minimal nonlinearity $\\alpha'(f,k)$ of a Boolean function $f$ can be when restricted to $k$-dimensional affine subspaces of $\\mathbb{F}_2^n$, defining $g(n,k)$ and $g'(n,k)$. It derives general bounds using combinatorial and probabilistic arguments (including a key criterion linking degree to flat-sums, and density arguments for low-degree/low-nonlinearity functions), proves exact values in several near-extremal cases, and provides constructions and algorithms yielding explicit examples where $\\alpha(f,k)$ or $\\alpha'(f,k)$ attains high values. The paper also tabulates computed values for small $n$ and $k$, identifies precise relationships between $g$ and $g'$, and puts forward a conjecture $g(k+2,k)=k-2$ with partial verification and heuristic support. Collectively, these results advance understanding of how Boolean functions behave under subspace restrictions, with implications for normality phenomena and potential cryptanalytic considerations under partial-domain reductions.
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.