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

2024-10-21

arXiv.org

AI Summary

This work extends the Leftover Hash Lemma (LHL) to *k*-universal hash functions and establishes unified uniformity guarantees via α-Rényi divergence (α ∈ (1, ∞]). Methodologically, it derives the first α-Rényi divergence bounds for *k*-universal hashing valid across the full range of α, leverages α-Rényi entropy to quantify source randomness, and employs conditional entropy modeling and probability distance analysis to handle settings with auxiliary information. The main contributions are threefold: (1) For α ≤ *k*, it achieves near-lossless extraction in terms of α-Rényi entropy, with tight, bounded α-Rényi divergence between the output distribution and the uniform distribution; (2) As *k* grows large, the extractor asymptotically attains optimal performance under min-entropy; (3) It generalizes these results to leakage-resilient hashing—i.e., hashing in the presence of side information—thereby significantly enhancing the robustness of randomness extraction in adversarial settings.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Natural Language Processing: Information Extraction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $2le lle k$. As a common distinctive feature, our results provide estimates of closeness to uniformity in terms of the $alpha$-R'enyi divergence for all $alphain (1,infty]$. For $1le alphale k$ we show that it is possible to convert all the randomness of the source measured in $alpha$-R'enyi 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.

Problem

• Extending uniformity guarantees to k-universal hash functions using Rényi divergence
• Converting source randomness into uniform bits across entropy measures
• Generalizing hashing results to scenarios with side information

Innovation

• Extends Leftover Hash Lemma to k*-universal hash functions
• Uses Rényi divergence for uniformity guarantees across α
• Enables randomness distillation with side information handling