Cryptology ePrint Archive: Report 2019/1035

An LLL Algorithm for Module Lattices

Changmin Lee and Alice Pellet-Mary and Damien Stehlé and Alexandre Wallet

Abstract: The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors. We provide a generalization to R-modules contained in K^n for arbitrary number fields K and dimension n, with R denoting the ring of integers of K. Concretely, we introduce an algorithm that efficiently finds short vectors in rank-n modules when given access to an oracle that finds short vectors in rank-2 modules, and an algorithm that efficiently finds short vectors in rank-2 modules given access to a Closest Vector Problem oracle for a lattice that depends only on K. The second algorithm relies on quantum computations and its analysis is heuristic. In the special case of free modules, we propose a dequantized version of this algorithm.

Category / Keywords: foundations /

Original Publication (with major differences): IACR-ASIACRYPT-2019

Date: received 11 Sep 2019, last revised 19 Jun 2020

Contact author: changmin lee at ens-lyon fr, alice pellet-mary at ens-lyon org, damien stehle at ens-lyon fr, wallet alexandre at gmail com

Available format(s): PDF | BibTeX Citation

Note: update: dequantizing the algorithm for free modules

Version: 20200619:141655 (All versions of this report)

Short URL:

[ Cryptology ePrint archive ]