R\'enyi divergence guarantees for hashing with linear codes

TL;DR

This paper analyzes how linear hashing, especially with Reed-Muller matrices, can effectively transform unknown sources into nearly uniform distributions, with guarantees provided by Re9nyi divergence estimates for all integer pb2.

Contribution

It provides the first estimation of expected p-divergence from uniform for random linear codes and demonstrates the effectiveness of Reed-Muller matrices in distribution smoothing.

Findings

Expected p-divergence bounds for random linear codes.

Reed-Muller matrices achieve intrinsic randomness in Bernoulli sources.

Distribution smoothing via linear codes is quantitatively characterized.

Abstract

We consider the problem of distilling uniform random bits from an unknown source with a given $p$-entropy using linear hashing. As our main result, we estimate the expected $p$-divergence from the uniform distribution over the ensemble of random linear codes for all integer $p\ge 2$. The proof relies on analyzing how additive noise, determined by a random element of the code from the ensemble, acts on the source distribution. This action leads to the transformation of the source distribution into an approximately uniform one, a process commonly referred to as distribution smoothing. We also show that hashing with Reed-Muller matrices reaches intrinsic randomness of memoryless Bernoulli sources in the $l_p$ sense for all integer $p\ge 2$.