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**2023/317**

**Title:**

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

**Authors:**
Razvan Barbulescu, Adrien Poulalion

**Abstract:**

Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is \tilde{O}(m^5). In this work we propose a modification of the algorithm for which the number of qubits is \tilde{O}(m^2) in the case of cyclotomic fields. Moreover, under a recent conjecture on the size of the class group of \mathbb{Q}(\zeta_m+\zeta_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 thnaks to the Buchmann-Pohst algorithm over \mathcal{O}_K-lattices.

**ePrint:**
https://eprint.iacr.org/2023/317

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