Permutation Superposition Oracles for Quantum Query Lower Bounds

TL;DR

This paper extends the compressed oracle method to random permutations, enabling new lower bounds in quantum query complexity, and proves the preimage resistance of the one-round sponge construction in this model.

Contribution

It introduces a generalized oracle simulation technique for random permutations, overcoming previous limitations and enabling new cryptographic security proofs.

Findings

Bound success probability of algorithms on permutations

Proved unconditional preimage resistance of the sponge construction

Generalized Zhandry's method to permutations

Abstract

We propose a generalization of Zhandry's compressed oracle method to random permutations, where an algorithm can query both the permutation and its inverse. We show how to use the resulting oracle simulation to bound the success probability of an algorithm for any predicate on input-output pairs, a key feature of Zhandry's technique that had hitherto resisted attempts at generalization to random permutations. One key technical ingredient is to use strictly monotone factorizations to represent the permutation in the oracle's database. As an application of our framework, we show that the one-round sponge construction is unconditionally preimage resistant in the random permutation model. This proves a conjecture by Unruh.