Keskeiset käsitteet
A randomly chosen linear map over a finite field gives a good hash function in the ℓ∞ sense, with the output being close to uniform in the ℓ∞ norm. This complements the widely-used Leftover Hash Lemma which provides guarantees in the ℓ1 norm.
Tiivistelmä
The main result of this paper is to show that a randomly chosen linear map over a finite field gives a good hash function in the ℓ∞ sense. Specifically:
Consider a set S ⊂ Fₙ_q and a randomly chosen linear map L: Fₙ_q → Fₜ_q, where qᵗ is sufficiently smaller than |S|.
The 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. This means that every element in the range Fₜ_q has about the same number of elements in S mapped to it.
This complements the Leftover Hash Lemma (LHL) which proves an analog statement under the ℓ₁ distance for a richer class of functions.
The 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.
The results are shown to be tight, demonstrating that linear functions hash as well as truly random functions up to a constant factor in the entropy loss.