An algorithm for computing the Stickelberger elements for imaginary multiquadratic fields

In this paper we present an algorithm for computing the Stickelberger ideal for multiquadratic fields $K=\mathbb{Q}(\sqrt{d_1}, \sqrt{d_2},\ldots,\sqrt{d_n})$, where $d_i \equiv 1 \pmod 4$ for $i=1,\ldots,n$ and $d_i$'s are pair-wise co-prime. Our result is based on the work of R. Kucera [J. Number Theory 56, 1996]. We systematize the ideas of this work, put them into explicit algorithms, prove their correctness and complexity. For $2^n = [K : \mathbb{Q}]$, our algorithm runs for time $\widetilde{\mathcal{O}}(2^n)$. We hope that the obtained results will serve as the first step towards solving the shortest vector problem for ideals of multiquadratic fields, which is the core problem in lattice-based cryptography.