Linear Hashing With ell_infty Guarantees And Two-sided Kakeya Bounds
Dhar Manik, Dvir Zeev. TheoretiCS Volume 2022
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.
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