Paper 2025/1797
An efficient quantum algorithm for computing $S$-units and its applications
Abstract
In this paper, we provide details on the proofs of the quantum polynomial time algorithm of Biasse and Song (SODA 16) for computing the $S$-unit group of a number field. This algorithm directly implies polynomial time methods to calculate class groups, $S$-class groups, relative class group and unit group, ray class groups, solve the principal ideal problem, solve certain norm equations, and decompose ideal classes in the ideal class group. Additionally, combined with a result of Cramer, Ducas, Peikert and Regev (Eurocrypt 2016), the resolution of the principal ideal problem allows one to find short generators of a principal ideal. Likewise, methods due to Cramer, Ducas and Wesolowski (Eurocrypt 2017) use the resolution of the principal ideal problem and the decomposition of ideal classes to find so-called ``mildly short vectors'' in ideal lattices of cyclotomic fields.
Note: Long version of a result published in the proceedings of SODA 2016.
Metadata
- Available format(s)
-
PDF
- Category
- Attacks and cryptanalysis
- Publication info
- Published elsewhere. Major revision. SODA 2016
- DOI
- 10.1137/1.9781611974331.ch64
- Keywords
- Ideal latticesQuantum AttacksS-unitsPrincipal Ideal ProblemHidden Subgroup Problem
- Contact author(s)
-
biasse @ usf edu
fang song @ pdx edu - History
- 2025-11-24: revised
- 2025-10-01: received
- See all versions
- Short URL
- https://ia.cr/2025/1797
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2025/1797,
author = {Jean-François Biasse and Fang Song},
title = {An efficient quantum algorithm for computing $S$-units and its applications},
howpublished = {Cryptology {ePrint} Archive, Paper 2025/1797},
year = {2025},
doi = {10.1137/1.9781611974331.ch64},
url = {https://eprint.iacr.org/2025/1797}
}
