Rényi divergence-based uniformity guarantees for k-Universal Hash Functions

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*-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 ≤ k we 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 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.