Explicit Nonlinear Functions beyond the Fourier bound

2026-10-05

AI Summary

This study addresses the long-standing challenge that constructing highly nonlinear vector functions over finite fields has been constrained by classical Fourier-analytic bounds. To overcome this limitation, the project establishes, for the first time, a theoretical connection between polynomial equation counting and nonlinear functions. By integrating techniques from algebraic geometry, combinatorics, hypergraph theory, and finite field arithmetic, it transcends traditional barriers through algebraic counting and graph-theoretic methods. The principal contributions include the successful construction of nonlinear mappings and low-degree polynomial interaction functions that surpass established Fourier bounds, thereby achieving exponentially superior nonlinearity measures. Furthermore, these results are generalized to arbitrary finite field settings, significantly advancing the theoretical foundations for designing cryptographic primitives with enhanced resistance to linear approximation attacks.

Abstract

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)>0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = \Theta(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $\gamma>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_{\gamma}(n)$, achieving $$ A(m,n) \leq (1 + \gamma)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{\gamma, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

Problem

• nonlinear functions
• vectorial maps
• Fourier bound
• affine agreement
• polynomial tuples

Innovation

• nonlinear functions
• Fourier bound
• polynomial equations
• hypergraphs
• finite fields