Faster Mixing of Higher-Dimensional Random Reversible Circuits

TL;DR

This paper introduces a new class of random reversible circuits with faster mixing times by utilizing higher-dimensional lattice architectures, improving over previous linear-in-n depth dependencies.

Contribution

It presents the first construction of higher-dimensional lattice-based random reversible circuits with sublinear depth dependence, inspired by cryptography and block cipher design.

Findings

Achieves sublinear-in-n depth for approximate k-wise independence

Introduces higher-dimensional lattice architecture for reversible circuits

Improves upon previous linear-in-n depth constructions

Abstract

We continue the study of the approximate $k$-wise independence of random reversible circuits as permutations of $\{\pm1\}^n$. Our main result is the first construction of a natural class of random reversible circuits with a sublinear-in-$n$ dependence on depth. Our construction is motivated by considerations in practical cryptography and is somewhat inspired by the design of practical block ciphers, such as DES and AES. Previous constructions of He and O'Donnell [HO24], which were built with gate architectures on one-dimensional lattices, suffered from an inherent linear-in-$n$ dependence on depth. The main novelty of our circuit model is a gate architecture built on higher-dimensional lattices.