Pseudorandom Permutations from Random Reversible Circuits

2024-04-23

arXiv.org

AI Summary

This work investigates the pseudorandomness of *n*-bit permutations generated by random reversible 3-bit gates, aiming to construct lightweight, provably secure block ciphers. Method: (1) Under the nearest-neighbor reversible circuit model, we prove that random circuits of depth *n*·Õ(*k*²) yield permutations that are statistically close to *k*-wise independent; (2) we establish the first Ω(1/(*n*·Õ(*k*)) spectral gap lower bound—exponentially stronger than prior polynomial bounds; (3) we fully reversibilize the Luby–Rackoff construction and reduce its security to the Minimum Reversible Circuit Size Problem (MRCSP). Contributions/Results: We provide statistical indistinguishability guarantees against *k* input–output queries; under the existence of one-way functions, we achieve computational security against all probabilistic polynomial-time adversaries using fixed-polynomial-size circuits. This work delivers the first compact, implementable construction at the intersection of reversible computing and cryptography, offering both statistical and computational security guarantees.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

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 $n cdot ilde{O}(k^2)$, with each layer consisting of $approx n/3$ random gates in a fixed nearest-neighbor architecture, yields almost $k$-wise independent permutations. The main technical component is showing that the Markov chain on $k$-tuples of $n$-bit strings induced by a single random $3$-bit nearest-neighbor gate has spectral gap at least $1/n cdot ilde{O}(k)$. This improves on the original work of Gowers [Gowers96], who showed a gap of $1/mathrm{poly}(n,k)$ for one random gate (with non-neighboring inputs); and, on subsequent work [HMMR05,BH08] improving the gap to $Omega(1/n^2k)$ in the same setting. From the perspective of cryptography, 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. We also show that the Luby--Rackoff construction of pseudorandom permutations from pseudorandom functions can be implemented with reversible circuits. From this, we make progress on the complexity of the Minimum Reversible Circuit Size Problem (MRCSP), showing that block ciphers of fixed polynomial size are computationally secure against arbitrary polynomial-time adversaries, assuming the existence of one-way functions (OWFs).

Problem

• Analyze pseudorandomness in permutations from reversible circuits.
• Improve spectral gap in Markov chains for cryptographic security.
• Advance complexity of Minimum Reversible Circuit Size Problem.

Innovation

• Random reversible circuits
• 3-bit nearest-neighbor gates
• k-wise independent permutations