Linear Hashing with $\ell_\infty$ guarantees and two-sided Kakeya bounds

TL;DR

This work establishes strong $\ell_\infty$-type uniformity guarantees for random linear hash functions over finite fields, showing that hashing a set $S \subset \mathbb{F}_q^n$ with a random linear map $L:\mathbb{F}_q^n\to\mathbb{F}_q^t$ yields $L(U_S)$ nearly uniform in $\ell_\infty$ with entropy loss close to optimal. The authors connect hashing with finite-field Kakeya/Furstenberg geometry, framing the problem in terms of $k$-dimensional shift-balanced subspaces and proving a two-sided Kakeya-type bound via a refined polynomial-method argument. They present formal theorems for large fields and the binary case, along with improved constants in the $\tau>1$ regime and a detailed treatment of the polynomial-method machinery (multiplicities, Hasse derivatives, and the EVAL/Coeff matrices). The results tightly characterize when linear hashing matches truly random behavior up to a constant factor in entropy loss and illuminate fundamental limits through lower bounds, tightness results, and connections to deep geometric-combinatorial problems. Practically, this yields simple, implementable linear hash families with strong per-bucket guarantees, with potential cryptographic and load-balancing applications that require precise control over worst-case bucket sizes.

Abstract

We show that a randomly chosen linear map over a finite field gives a good hash function in the $\ell_\infty$ sense. More concretely, consider a set $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to \mathbb{F}_q^t$ with $q^t$ taken to be sufficiently smaller than $ |S|$. Let $U_S$ denote a random variable distributed uniformly on $S$. Our main theorem shows that, with high probability over the choice of $L$, the random variable $L(U_S)$ is close to uniform in the $\ell_\infty$ norm. In other words, {\em every} element in the range $\mathbb{F}_q^t$ has about the same number of elements in $S$ mapped to it. This complements the widely-used Leftover Hash Lemma (LHL) which proves the analog statement under the statistical, or $\ell_1$, distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [ADMPT99]. By known bounds from the load balancing literature [RS98], our results are tight and show that linear functions hash as well as trully random function up to a constant factor in the entropy loss. Our proof leverages a connection between linear hashing and the finite field Kakeya problem and extends some of the tools developed in this area, in particular the polynomial method.

Theorems & Definitions (35)

• Lemma 1.1: Leftover Hash Lemma ILL
• Theorem 1.2: Main theorem (informal)
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• definition 3.1