Pseudorandomness Properties of Random Reversible Circuits

2025-02-11

AI Summary

This work addresses the provable security of pseudorandom permutations in lightweight cryptography, specifically investigating whether shallow random reversible circuits—composed of 3-bit gates on a 2D nearest-neighbor architecture—can achieve *k*-wise independence. We propose an alternating row/column permutation construction and combine *k*-tuple permutation modeling with spectral analysis of Markov chains. For the first time under this architecture, we prove that random reversible circuits of depth *O*(√*n* · *k*³ polylog(*n*, *k*)) suffice to generate permutations over {0,1}ⁿ that are statistically close to *k*-wise independent. A key innovation is improving the spectral gap lower bound to Ω(1/(*n* · Õ(*k*))), substantially surpassing prior polynomial-gap results. Our construction yields a low-depth, hardware-friendly block cipher prototype with rigorous statistical security guarantees—specifically, provable resistance against *k*-query plaintext–ciphertext attacks.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Motivated by practical concerns in cryptography, we study pseudorandomness properties of permutations on ${0,1}^n$ computed by random circuits made from reversible $3$-bit gates (permutations on ${0,1}^3$). Our main result is that a random circuit of depth $sqrt{n} cdot ilde{O}(k^3)$, with each layer consisting of $Theta(n)$ random gates in a fixed two-dimensional nearest-neighbor architecture, yields approximate $k$-wise independent permutations. Our result can be seen as a particularly simple/practical block cipher construction that gives provable statistical security against attackers with access to $k$~input-output pairs within few rounds. The main technical component of our proof consists of two parts: 1. We show that the Markov chain on $k$-tuples of $n$-bit strings induced by a single random $3$-bit one-dimensional nearest-neighbor gate has spectral gap at least $1/n cdot ilde{O}(k)$. Then we infer that a random circuit with layers of random gates in a fixed one-dimensional gate architecture yields approximate $k$-wise independent permutations of ${0,1}^n$ in depth $ncdot ilde{O}(k^2)$ 2. We show that if the $n$ wires are layed out on a two-dimensional lattice of bits, then repeatedly alternating applications of approximate $k$-wise independent permutations of ${0,1}^{sqrt n}$ to the rows and columns of the lattice yields an approximate $k$-wise independent permutation of ${0,1}^n$ in small depth. Our work improves on the original work of Gowers, who showed a gap of $1/mathrm{poly}(n,k)$ for one random gate (with non-neighboring inputs); and, on subsequent work improving the gap to $Omega(1/n^2k)$ in the same setting.

Problem

• Study pseudorandomness in reversible circuits
• Prove security against k input-output pairs
• Improve Gowers' spectral gap results

Innovation

• Random reversible 3-bit gates
• Two-dimensional nearest-neighbor architecture
• Approximate k-wise independent permutations