The Special Case of Cyclotomic Fields in Quantum Algorithms for Unit Groups

Abstract

Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is O~ (m5). In this work we propose a modification of the algorithm for which the number of qubits is O~ (m2) in the case of cyclotomic fields. Moreover, under a recent conjecture on the size of the class group of Q(ζm+ζm-1), the quantum algorithm is much simpler because it is a hidden subgroup problem (HSP) algorithm rather than its error estimation counterpart: continuous hidden subgroup problem (CHSP). We also discuss the (minor) speed-up obtained when exploiting Galois automorphisms thanks to the Buchmann-Pohst algorithm over OK -lattices.