Linear Hashing with l∞ guarantees and two-sided Kakeya bounds

We show that a randomly chosen linear map over a finite field gives a good hash function in the l(infinity) sense. More concretely, consider a set S subset of F-q(n) and a randomly chosen linear map L : F-q(n) -> 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 l(infinity) norm. In other words, every element in the range 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 l(1), distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [1]. By known bounds from the load balancing literature [2], our results are tight and show that linear functions hash as well as truly 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.