超越傅里叶界的显式非线性函数

中文摘要 AI 辅助

我们研究构造高度非线性向量映射 $F: \mathbb F_2^n \to \mathbb F_2^m$ 的问题。具体而言，我们希望找到一个 $F$ 以及尽可能小的 $A = A(m,n)> 0$，使得对于每个仿射映射 $L: \mathbb F_2^n \to \mathbb F_2^m$（形如 $L(x) = M x + b$），我们有：$$\mathrm{agree}(F, L):= |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ 此类问题已被 Nyberg (1991,1993)、Carlet 和 Ding (2004,2007)、Liu、Mesnager 和 Chen (2017)、Nagy (2025) 以及 Biryukov、Turecek 和 Udovenko (2026) 研究过。有一种经典方法利用 bent 函数和傅里叶分析思想来构造此类函数；该方法能达到的最佳界为：$$ A(m,n) = \Theta(2^{n-m} + 2^{n/2}),$$ 特别是，该值从不小于 $2^{n/2}$。在本工作中，我们展示了如何构造超越此傅里叶界的高度非线性函数。具体而言，我们展示了如何为每个 $\gamma>0$ 构造一个函数 $F: \mathbb F_2^n \to \mathbb F_2^m$，其中 $m = O_{\gamma}(n)$，并达到 $$ A(m,n) \leq (1 + \gamma)^n.$$ 令人惊讶的是，对于更难的问题——即与 $m$ 元组 $d$ 次多项式 $Q: \mathbb F_2^n \to \mathbb F_2^m$ 具有低一致度的问题（其中 $m = O_{\gamma, d}(n)$），我们甚至达到了相同的定量行为。此前在此问题上的最佳界是 Ben-Sasson 和 Kopparty (2010) 基于 Gowers 范数类型论证得到的 $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$。我们的所有结果都推广到所有有限域 $\mathbb F_q$ 以替代 $\mathbb F_2$。我们的方法基于与经典代数方法计数多项式方程组解的结果的新联系。这一联系将我们引向组合学中的基本问题，即关于同时具有少量边和独立集的图和超图的问题。

英文摘要

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: $$ A(m,n) = Θ(2^{n-m} + 2^{n/2}),$$ 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 $γ>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_γ(n)$, achieving $$ A(m,n) \leq (1 + γ)^n.$$ 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 $m = O_{γ, d}(n)$. 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.

发表机构

• University of Toronto（多伦多大学）
• Tata Institute of Fundamental Research, Mumbai（孟买塔塔基础研究所）

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29 pages