We show that a randomly chosen linear map over a finite field gives a good
hash function in the βββ sense. More concretely, consider a set SβFqnβ and a randomly chosen linear map L:FqnββFqtβ with qt taken to be sufficiently smaller than β£Sβ£. Let
USβ 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(USβ) is close to uniform in the βββ norm. In other words, every
element in the range Fqtβ 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 β1β, distance
(for a richer class of functions) as well as prior work on the expected largest
'bucket size' in linear hash functions [ADMPT99]. 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