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 $m=2^{t}p$, 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 $$, which can then be tackled in polynomial time, by combining methods by Friedl et al. for $$-torsion groups and by Bonnetain and Naya-Plasencia for $$-torsion groups.