Jean-Francois Biasse | Searcharxiv

arXiv2025

An efficient quantum algorithm for computing $S$-units and its applications

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.

cs.CR↗

arXiv2017

A fast algorithm for finding a short generator of a principal ideal of $\mathbb{Q}(ζ_{p^s})$

We present a heuristic algorithm to compute the ideal class group, and a generator of a principal ideal in $\mathbb{Q}(ζ_{p^s})$ in time $2^{O(n^{1/2+\varepsilon})}$ for $n:= deg(K)$ and arbitrarily small $\varepsilon$. This yields an attack on the schemes relying on the hardness of finding a short generator of a principal ideal such as such as the homomorphic encryption scheme of Vercauteren and Smart, and the multilinear maps of Garg, Gentry and Halevi. We rely on the work from Cramer, Ducas, Peikert and Regev. They proved that finding a short generator polynomially reduces to finding an arbitrary one. The complexity is better than when we rely on the work of Biasse and Fieker on the PIP, which yields an attack in time $2^{n^{2/3+\varepsilon}}$ for arbitrarily small $\varepsilon >0$. $\textbf{Since Sep. 30 2016}$ We present practical improvements to our methods. Moreover, we describe a variant that solves the PIP on input ideal $I$ of norm less than $2^{n^b}$ in time $2^{O\left(n^{c+o(1)}\right)}$ for $2/5 < c < 1/2$ and $b\leq 7c -2$ given a one time precomputation of cost $2^{O(n^{2-3c+\varepsilon})}$ for an arbitrarily small $\varepsilon$. This also solves $γ$-SVP in principal ideals of $\mathbb{Q}(ζ_{p^s})$ for $γ\in e^{\tilde{O}(\sqrt{n})}$. On principal ideals of norm less than $2^{n^b}$, we can leverage the precomputation to achieve a better asymptotic run time than the BKZ algorithm.

math.NT↗