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 $k^\ast$-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 $\alpha$-R\'enyi divergence for all $\alpha\in (1,\infty]$. For $1\le \alpha\le k$ 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.