Year: 2025
Type: article
Abstract: Universal hash functions map the output of a source to random strings over a finite alphabet, aiming to approximate the uniform distribution on the set of strings. A classic result on these functions, called the Leftover Hash Lemma, gives an estimate of the distance from uniformity based on the assumptions about the min-entropy of the source. We prove several results concerning extensions of this lemma to a class of functions that arek*-universal, i.e.,l-universal for all 2 ≤l≤k. As a common distinctive feature, our results provide estimates of closeness to uniformity in terms of the α-Rényi divergence for all α ∈ (1, ∞]. For 1 ≤ α ≤kwe show that it is possible to convert all the randomness of the source measured in α-Rényi entropy into approximately uniform bits with nearly the same amount of randomness. For large enoughkwe show that it is possible to distill random bits that are nearly uniform, as measured by min-entropy. We also extend these results to hashing with side information.
Authors: Madhura Pathegama, Alexander Barg
Institution: University of Maryland, College Park
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4415003199