Paper 2022/1661

Enhancing the Dual Attack against MLWE: Constructing More Short Vectors Using Its Algebraic Structure

Abstract

Primal attack, BKW attack, and dual attack are three well-known attacks to LWE. To build efficient post-quantum cryptosystems in practice, the structured variants of LWE (i.e. MLWE/RLWE) are often used. Some efforts have been spent on addressing concerns about additional vulnerabilities introduced by algebraic structures and no effective attack method based on ideal lattices or module lattices has been proposed so far; these include refining primal attack and BKW attack to MLWE/RLWE. It is thus an interesting problem to consider how to enhance the dual attack against LWE with the rich algebraic structure of MLWE (including RLWE). In this paper, we present the first attempt to this problem by observing that each short vector found by BKZ generates another n − 1 vectors of the same length automatically and all of these short vectors can be used to distinguish. To this end, an interesting property which indicates the rotations are consistent with certain linear transformations is proved, and a new kind of intersection lattice is constructed with some tricks. Moreover, we notice that coefficient vectors of different rotations of the same polynomial are near-orthogonal in high-dimensional spaces. This is validated by extensive experiments and is treated as an extension to the assumption under the original dual attack against LWE. Taking Newhope512 as an example, we show that by our enhanced dual attack method, the required blocksize and time complexity (in both classical and quantum cases) all decrease. It is remarked that our improvement is not significant and its limitation is also touched on. Our results do not reveal a severe security problem for MLWE/RLWE compared to that of a general LWE, this is consistent with the findings by the previous work for using primal and BKW attacks to MLWE/RLWE.