The special case of cyclotomic fields in quantum algorithms for unit groups

03/07/2023

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by   Razvan Barbulescu, et al.

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Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is Õ(m^5). In this work we propose a modification of the algorithm for which the number of qubits is Õ(m^2) in the case of cyclotomic fields. Moreover, under a recent conjecture on the size of the class group of ℚ(ζ_m + ζ_m^-1), the quantum algorithms 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 𝒪_K-lattices.