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Rényi divergence-based uniformity guarantees for $k$-universal hash functions

Madhura Pathegama, Alexander Barg

TL;DR

The paper advances uniformity guarantees for $k^*$-universal hash functions by deriving one-shot, non-asymptotic bounds based on $α$-Rényi divergences for $α∈(1,k]$, showing that nearly all the $H_α(X)$ entropy of a source can be distilled into output strings that are close to uniform. It further extends these guarantees to α>k via conditional Rényi divergences, providing strong min-entropy–based privacy amplification results that reduce dependence on the seed. A complete treatment is given for side information, yielding analogous LHL-type results with $H_α(X|Z)$, and a detailed analysis of the largest hash bucket to assess practical hash-performance implications. The work highlights seed-length considerations and points to directions for shorter-seed extractors and extensions to almost-$k^*$-universal families, with clear cryptographic relevance for secrecy and key-generation applications.

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 are $k^\ast$-universal, i.e., $l$-universal for all $2\le l\le k$. As a common distinctive feature, our results provide estimates of closeness to uniformity in terms of the $α$-R{é}nyi divergence for all $α\in (1,\infty]$. For $1\le α\le k$ we show that it is possible to convert all the randomness of the source measured in $α$-\Renyi entropy into approximately uniform bits with nearly the same amount of randomness. For large enough $k$ we 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.

Rényi divergence-based uniformity guarantees for $k$-universal hash functions

TL;DR

The paper advances uniformity guarantees for -universal hash functions by deriving one-shot, non-asymptotic bounds based on -Rényi divergences for , showing that nearly all the entropy of a source can be distilled into output strings that are close to uniform. It further extends these guarantees to α>k via conditional Rényi divergences, providing strong min-entropy–based privacy amplification results that reduce dependence on the seed. A complete treatment is given for side information, yielding analogous LHL-type results with , and a detailed analysis of the largest hash bucket to assess practical hash-performance implications. The work highlights seed-length considerations and points to directions for shorter-seed extractors and extensions to almost--universal families, with clear cryptographic relevance for secrecy and key-generation applications.

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 are -universal, i.e., -universal for all . As a common distinctive feature, our results provide estimates of closeness to uniformity in terms of the -R{é}nyi divergence for all . For we show that it is possible to convert all the randomness of the source measured in -\Renyi entropy into approximately uniform bits with nearly the same amount of randomness. For large enough we 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.

Paper Structure

This paper contains 11 sections, 16 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Measures of randomness
  4. Proximity measures for distributions
  5. Hash functions
  6. --universality and uniformity guarantees
  7. Distilling min-entropy
  8. Largest hash bucket
  9. Leftover hashing with side information
  10. Concluding remarks
  11. Proof of Theorem \ref{['thm: main2']}

Key Result

Proposition 2.1

Let $X$ be a random variable defined on ${\EuScript X}$ and let $S$ be a uniform random variable $S \sim {\EuScript S}$ that is independent of $X$. Let $h : {\EuScript S} \times {\EuScript X} \to {\mathbb Z}_q^m$ be a universal hash function. If $m \leq H_{\infty}(X)-\log_q(1/\epsilon)$, then

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1: Leftover hash lemma impagliazzo1989pseudo
  • Proposition 2.2
  • Remark 1
  • Definition 2.4
  • Proposition 2.3: Renner2008 Corollary 5.6.1
  • Theorem 3.1
  • Theorem 3.2
  • ...and 21 more