**A fusion algorithm for solving the hidden shift problem in finite abelian groups**

*Wouter Castryck and Ann Dooms and Carlo Emerencia and Alexander Lemmens*

**Abstract: **It follows from a result by Friedl, Ivanyos, Magniez, Santha and Sen from 2014 that, for any fixed integer $m > 0$ (thought of as being small), there exists a quantum algorithm for solving the hidden shift problem in an arbitrary finite abelian group $(G, +)$ with time complexity poly$( \log |G|) \cdot 2^{O(\sqrt{\log |mG|})}$. As discussed in the current paper, this can be viewed as a modest statement of Pohlig-Hellman type for hard homogeneous spaces. Our main contribution is a somewhat simpler algorithm achieving the same runtime for $m = 2^tp$, with $t$ any non-negative integer and $p$ any prime number, where additionally the memory requirements are mostly in terms of quantum random access classical memory; indeed, the amount of qubits that need to be stored is poly$( \log |G|)$. Our central tool is an extension of Peikert's adaptation of Kuperberg's collimation sieve to arbitrary finite abelian groups. This allows for a reduction, in said time, to the hidden shift problem in the quotient $G/2^tpG$, which can then be tackled in polynomial time, by combining methods by Friedl et al. for $p$-torsion groups and by Bonnetain and Naya-Plasencia for $2^t$-torsion groups.

**Category / Keywords: **public-key cryptography / hidden shift, collimation sieve, hard homogeneous space

**Date: **received 28 Apr 2021, last revised 27 May 2021

**Contact author: **wouter castryck at esat kuleuven be, ann dooms at vub be, carlo emerencia at vub be, alexander lemmens at esat kuleuven be

**Available format(s): **PDF | BibTeX Citation

**Version: **20210527:071403 (All versions of this report)

**Short URL: **ia.cr/2021/562

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