More Efficient $k$-wise Independent Permutations from Random Reversible Circuits via log-Sobolev Inequalities

TL;DR

This paper demonstrates that reversible circuits with a specific number of random 3-bit gates can efficiently produce permutations that are approximately k-wise independent, improving bounds in certain regimes by analyzing log-Sobolev constants.

Contribution

It introduces a new analysis method using log-Sobolev inequalities to bound the efficiency of generating k-wise independent permutations with reversible circuits.

Findings

Reversible circuits with nk log(1/psilon)" gates achieve psilon-approximate k-wise independence.

Analysis based on log-Sobolev constants provides improved bounds over spectral gap methods.

The results apply to the regime where the approximation error psilon is not too small.

Abstract

We prove that the permutation computed by a reversible circuit with $\tilde{O}(nk\cdot \log(1/\varepsilon))$ random $3$-bit gates is $\varepsilon$-approximately $k$-wise independent. Our bound improves on currently known bounds in the regime when the approximation error $\varepsilon$ is not too small. We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.