Efficient Construction of Pseudorandom Permutations from Random Reversible Circuits

Core Concepts

Random reversible circuits with a simple brickwork architecture can efficiently compute almost k-wise independent permutations, providing provable statistical security against attackers with access to k input-output pairs.

Abstract

The paper studies pseudorandomness properties of permutations computed by small, randomly chosen reversible circuits. The main results show that:

Random reversible circuits with a simple brickwork architecture can compute almost k-wise independent permutations, providing provable statistical security against attackers with access to k input-output pairs. Specifically, a random brickwork circuit of depth n·e^(O(k^2)) can yield 2^(-O(nk))-approximate k-wise independent permutations.

This improves on previous work, which required more complex circuit architectures and larger depths to achieve similar levels of pseudorandomness.

The key technical component is proving that the Markov chain on k-tuples of n-bit strings induced by a single random 3-bit nearest-neighbor gate has a spectral gap of at least 1/n·e^(O(k)). This is a significant improvement over prior work.

The paper also shows that the Luby-Rackoff construction of pseudorandom permutations from pseudorandom functions can be implemented with reversible circuits, making progress on the complexity of the Minimum Reversible Circuit Size Problem (MRCSP).

Assuming the existence of one-way functions, the paper establishes that block ciphers of fixed polynomial size are computationally secure against arbitrary polynomial-time adversaries, even if the adversary has access to a polynomial number of input-output pairs.

Stats

A random brickwork circuit of depth n·e^(O(k^2)) can yield 2^(-O(nk))-approximate k-wise independent permutations.
The Markov chain on k-tuples of n-bit strings induced by a single random 3-bit nearest-neighbor gate has a spectral gap of at least 1/n·e^(O(k)).
Assuming one-way functions exist, block ciphers of fixed polynomial size are computationally secure against arbitrary polynomial-time adversaries, even if the adversary has access to a polynomial number of input-output pairs.

Quotes

"Our main result is that a random circuit of depth n · e^O(k^2), with each layer consisting of ≈n/3 random gates in a fixed nearest-neighbor architecture, yields almost k-wise independent permutations."
"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."

Key Insights Distilled From

Pseudorandom Permutations from Random Reversible Circuits

by William He,R... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14648.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to analyze the pseudorandomness of quantum circuits or other models of reversible computation

The techniques developed in this paper for analyzing the pseudorandomness of circuits with random reversible gates can be extended to analyze the pseudorandomness of quantum circuits. Quantum circuits are composed of quantum gates that operate on qubits, and the principles of reversible computation apply to quantum computation as well. By adapting the spectral analysis and comparison methods used in this paper to the context of quantum circuits, researchers can investigate the pseudorandomness properties of quantum permutations and quantum operations. This can be particularly useful in quantum cryptography and quantum information processing, where ensuring the security and randomness of quantum operations is crucial.

What are the implications of the hardness result for the Minimum Reversible Circuit Size Problem (MRCSP) on the design and analysis of practical block ciphers

The hardness result for the Minimum Reversible Circuit Size Problem (MRCSP) has significant implications for the design and analysis of practical block ciphers. The MRCSP is a fundamental problem in cryptography that deals with determining the minimum size of a reversible circuit that can compute a given permutation. The result showing the hardness of MRCSP under the existence of one-way functions implies that finding the minimum reversible circuit size for a given permutation is computationally challenging. This has implications for the security of block ciphers, as it suggests that designing efficient and secure block ciphers based on reversible circuits is a complex task that requires careful consideration of computational complexity and cryptographic assumptions. The result underscores the importance of using strong cryptographic primitives and techniques in the design of secure block ciphers.

Can the insights from this work on spectral gaps of Markov chains induced by random reversible gates be applied to study the mixing time of other Markov processes in computer science and physics

The insights from this work on spectral gaps of Markov chains induced by random reversible gates can be applied to study the mixing time of other Markov processes in computer science and physics. The analysis of spectral gaps in Markov chains is a powerful tool for understanding the convergence properties of random processes and the efficiency of random walks on graphs. By applying similar techniques to different Markov chains in various domains, researchers can analyze the mixing time, convergence rate, and pseudorandomness properties of different stochastic processes. This can have applications in algorithm design, network analysis, statistical physics, and other fields where understanding the behavior of random processes is essential. The methods developed in this work can provide valuable insights into the spectral properties of diverse Markov chains and their implications for computational and physical systems.