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∞ sense. More concretely, consider a set S ⊂ Fq and a randomly chosen linear map L : Fq → Fq with q taken to be sufficiently smaller than |S|. Let US 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(US) is close to uniform in the l∞ norm. In other words, every element in the range Fq 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 l1, distance (for a richer class of functions) as well as prior work on the expected largest ’bucket size’ in linear hash functions [ADM99]. 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.