Paper 2026/1591

A Polynomial-Time Quantum Algorithm for the Dihedral Coset Problem

Abstract

We present a polynomial-time quantum algorithm for the Dihedral Coset Problem (DCP). The algorithm is based on Regev's polynomial-time reduction of the Dihedral Subgroup Problem (DSP) to the modular subset sum problem, but uses a different technique to erase sample bits without use of a subset sum oracle. The algorithm can thus combine with Regev's reduction of lattice problems to DCP, improved by Brakerski, Kirshanova, Stehl{\'e} and Wen, to yield polynomial-time quantum algorithms for various lattice problems, such as finding a polynomial-factor approximation to the shortest vector in an $n$-dimensional lattice (SVP), and the ``learning with errors'' problem (LWE). The algorithm can tolerate a faulty sample rate as high as $1/O(\log{n})$, allowing the algorithm-reduction combination to efficiently solve, for example, SVP with a $\sqrt{n}$ polylog($n$) approximation factor, or LWE instances with $\alpha=\sqrt{n}$ polylog($n$). Note: The proof of Lemma 3 in the first draft incorrectly conflated pairwise independence over the uniform vs. actual distribution on D. The proof has been substantially updated, and now distinguishes clearly between them. Additional note: A new preprint, https://eprint.iacr.org/2026/1693, has been posted claiming to prove that the algorithm in this paper can't possibly work. We're in the process of evaluating it. Additional note October 7/26: The preprint has been updated with a note addressing the claims in https://eprint.iacr.org/2026/1693, as well as clarifications to the proof of Lemma 3 in response to comments by Sam Jaques. Additional note November 3/26: The note mentioned above has been updated with edits for greater clarity. The paper has also been reformatted.