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

TL;DR

The paper shows that a random reversible circuit with $\tilde{O}(nk\log(1/\varepsilon))$ width-$2$ gates computes an $\varepsilon$-approximate $k$-wise independent permutation for $n$ and $k$ up to $2^{n/50}$. It achieves this by bounding log-Sobolev constants of associated Markov chains and transferring the bounds through a sequence of comparisons: uniform clique-coloring, standard clique-coloring, and the reversible-circuits walk, with a key generic-states refinement that yields a faster mixing time. The approach improves prior bounds by exploiting log-Sobolev inequalities (instead of spectral gaps) and a martingale-based analysis (Salez), culminating in a mixing time of $\tilde{O}(nk\log(1/\varepsilon))$. This provides a more efficient construction of pseudorandom permutations from shallow random reversible circuits, with implications for derandomization and cryptographic applications.

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.

Theorems & Definitions (42)

• Definition 1: Approximate $k$-wise independent permutations
• Theorem 1
• Definition 2: Tuples with distinct elements
• Definition 3: Width-$2$ simple permutations
• Definition 4: Mixing time
• Definition 5: Reversible Markov chain
• Definition 6: Dirichlet form
• Definition 7: Entropy
• Definition 8: Log-Sobolev constant of Markov chain
• Theorem 9: DSC96, Theorem 3.7