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 $n \cdot \tilde{O}(k^2)$, 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 $1/n \cdot \tilde{O}(k)$. 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).

翻译：我们研究了由可逆3比特门（在{0,1}³上的置换）构成的随机电路所计算的{0,1}ⁿ上置换的伪随机性性质。我们的主要结果表明：在固定最近邻架构中，每层包含约n/3个随机门、深度为n·Õ(k²)的随机电路，可产生近似k阶独立的置换。主要技术贡献在于证明：单个随机3比特最近邻门在n比特字符串的k元组上诱导的马尔可夫链，其谱间隙至少为1/n·Õ(k)。这改进了Gowers [Gowers96]的原始工作（其对单个随机门证明了1/poly(n,k)的间隙，且输入非相邻）；也改进了后续研究[HMMR05,BH08]（其在相同设定下将间隙提升至Ω(1/n²k)）。从密码学视角看，我们的结果可视为一种特别简洁/实用的分组密码构造，其在少量轮次内能为攻击者提供最多k个输入-输出对时，可证明的统计安全性。我们还证明了：基于伪随机函数构造伪随机置换的Luby–Rackoff方案，可通过可逆电路实现。由此，我们在最小可逆电路规模问题（MRCSP）的复杂度研究上取得进展，证明：若存在单向函数（OWF），则固定多项式规模的分组密码在计算上能抵御任意多项式时间攻击者。