Geodesic
An algorithm for computing the Stickelberger elements for imaginary multiquadratic fields
Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 12-17

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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.
Keywords: multiquadratic number field, Stickelberger ideal, Stickelberger element, the shortest vector problem. 
@article{PDMA_2020_13_a2,
     author = {D. O. Olefirenko and E. A. Kirshanova and E. S. Malygina and S. A. Novoselov},
     title = {An algorithm for computing the {Stickelberger} elements for imaginary multiquadratic fields},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {12--17},
     publisher = {mathdoc},
     number = {13},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2020_13_a2/}
}
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D. O. Olefirenko; E. A. Kirshanova; E. S. Malygina; S. A. Novoselov. An algorithm for computing the Stickelberger elements for imaginary multiquadratic fields. Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 12-17. http://geodesic.mathdoc.fr/item/PDMA_2020_13_a2/