Pseudorandomness Properties of Random Reversible Circuits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910822050037760 |
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| author | Gay, William He, William Kocurek, Nicholas O'Donnell, Ryan |
| author_facet | Gay, William He, William Kocurek, Nicholas O'Donnell, Ryan |
| contents | 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 $\sqrt{n} \cdot \tilde{O}(k^3)$, with each layer consisting of $Θ(n)$ 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 $1/n \cdot \tilde{O}(k)$. 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 $n\cdot \tilde{O}(k^2)$
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 $Ω(1/n^2k)$ in the same setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudorandomness Properties of Random Reversible Circuits Gay, William He, William Kocurek, Nicholas O'Donnell, Ryan Cryptography and Security Probability 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 $\sqrt{n} \cdot \tilde{O}(k^3)$, with each layer consisting of $Θ(n)$ 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 $1/n \cdot \tilde{O}(k)$. 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 $n\cdot \tilde{O}(k^2)$ 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 $Ω(1/n^2k)$ in the same setting. |
| title | Pseudorandomness Properties of Random Reversible Circuits |
| topic | Cryptography and Security Probability |
| url | https://arxiv.org/abs/2502.07159 |