Paper 2021/149

Quantum Security of the Legendre PRF

Paul Frixons and André Schrottenloher

Abstract

In this paper, we study the security of the Legendre PRF against quantum attackers, given classical queries only, and without quantum random-access memories. We give two algorithms that recover the key of a shifted Legendre symbol with unknown shift, with a complexity smaller than exhaustive search of the key. The first one is a quantum variant of the table-based collision algorithm. The second one uses Kuperberg's abelian hidden shift algorithm in an offline manner. We show that the latter, although asymptotically promising, is not currently the most efficient against practical parameters.