  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.
