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.

翻译：本文详细阐述了Biasse和Song（SODA 16）提出的用于计算数域$S$单位群的量子多项式时间算法的证明细节。该算法直接蕴含了计算类群、$S$类群、相对类群与单位群、射线类群，解决主理想问题，求解特定范数方程，以及在理想类群中分解理想类的多项式时间方法。此外，结合Cramer、Ducas、Peikert和Regev（Eurocrypt 2016）的结果，主理想问题的解决使得寻找主理想的短生成元成为可能。类似地，Cramer、Ducas和Wesolowski（Eurocrypt 2017）提出的方法利用主理想问题的解决和理想类的分解，在分圆域的理想格中寻找所谓的“适度短向量”。