Parallel Kac's Walk Generates PRU
Chuhan Lu, Minglong Qin, Fang Song, Penghui Yao, Mingnan Zhao
TL;DR
The paper proves that a parallel Kac's walk, repeated linearly many times, yields an adaptive-secure pseudorandom unitary (PRU) and attains strong security against inverse queries. It introduces a path-recording framework and a purified function-permutation oracle (HPO) to relate the Kac-based construction to Haar randomness, via a Compress isometry and a distinct-block subspace projection. The results establish that HP$_{n,T+1}$ is computationally indistinguishable from Haar, and HP$_{n,2T+1}$ achieves statistical strong-PRU security against adaptive and inverse-query attackers. This provides an alternative PRU construction and broadens evidence for the path-recording technique, with discussions on potential simplifications and open questions about round reduction and local circuit variants.
Abstract
Ma and Huang recently proved that the PFC construction, introduced by Metger, Poremba, Sinha and Yuen [MPSY24], gives an adaptive-secure pseudorandom unitary family PRU. Their proof developed a new path recording technique [MH24]. In this work, we show that a linear number of sequential repetitions of the parallel Kac's Walk, introduced by Lu, Qin, Song, Yao and Zhao [LQSY+24], also forms an adaptive-secure PRU, confirming a conjecture therein. Moreover, it additionally satisfies strong security against adversaries making inverse queries. This gives an alternative PRU construction, and provides another instance demonstrating the power of the path recording technique. We also discuss some further simplifications and implications.