Paper 2021/562

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

Wouter Castryck, Ann Dooms, 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.