Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.02280 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914169151815680 |
|---|---|
| author | Biasse, Jean-Francois Song, Fang |
| author_facet | Biasse, Jean-Francois Song, Fang |
| contents | 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 the 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02280 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An efficient quantum algorithm for computing $S$-units and its applications Biasse, Jean-Francois Song, Fang Cryptography and Security Number Theory 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 the 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. |
| title | An efficient quantum algorithm for computing $S$-units and its applications |
| topic | Cryptography and Security Number Theory |
| url | https://arxiv.org/abs/2510.02280 |