Faster Mixing of Higher-Dimensional Random Reversible Circuits

2024-09-22

arXiv.org

AI Summary

This work addresses the fundamental bottleneck in constructing random reversible circuits that approximate $k$-wise independence over ${pm1}^n$: existing approaches suffer from $Omega(n)$ depth dependence on dimension $n$. We propose the first reversible gate architecture grounded in high-dimensional lattice structures. Departing from conventional one-dimensional constructions—whose depth lower bound is inherently linear in $n$—we integrate reversible computation principles, permutation group theory, and block-cipher-inspired local interaction mechanisms to design a gate model featuring high-dimensional local coupling. This yields a natural family of random reversible circuits with depth $o(n)$, achieving the first sublinear dependence of mixing depth on $n$. Consequently, state-space mixing over high-dimensional domains is significantly accelerated. Our framework establishes a new paradigm for constructing approximate independence and reversible pseudorandomness, bridging structural combinatorics, group-theoretic analysis, and circuit-level design.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Faster mixing random reversible circuits
• Sublinear depth dependent on dimension
• Inspired by practical cryptography designs

Innovation

• Higher-dimensional gate architecture
• Sublinear depth dependence
• Inspired by block ciphers