Linear Hashing with ${\ell }_{\infty }$ guarantees and two-sided Kakeya bounds

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.

Journal Version for TheoretiCS. Added Theorem 3.4 which gives more flexible field size requirements for finding balanced subspaces