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- W4313203320 abstract "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 $Ssubsetmathbb{F}_{q}^{n}$ and a randomly chosen linear ${map}L:mathbb{F}_{q}^{n}rightarrowmathbb{F}_{q}^{t}$ with q <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>t</sup> taken to be sufficiently smaller than $|S|$. Let U <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>S</inf> 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, 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 [1]. By known bounds from the load balancing literature [2], our results are tight and show that linear functions hash as well as truly 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." @default.
- W4313203320 created "2023-01-06" @default.
- W4313203320 creator A5013053155 @default.
- W4313203320 creator A5034730963 @default.
- W4313203320 date "2022-10-01" @default.
- W4313203320 modified "2023-10-16" @default.
- W4313203320 title "Linear Hashing with ℓ<sub>∞</sub> guarantees and two-sided Kakeya bounds" @default.
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- W4313203320 doi "https://doi.org/10.1109/focs54457.2022.00047" @default.
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