Reductions from Module Lattices to Free Module Lattices, and Application to Dequantizing Module-LLL

In this article, we give evidence that free modules (i.e., modules which admit a basis) are no weaker than arbitrary modules, when it comes to solving cryptographic algorithmic problems (and when the rank of the module is at least 2). More precisely, we show that for three algorithmic problems used in cryptography, namely the shortest vector problem, the Hermite shortest vector problem and a variant of the closest vector problem, there is a reduction from solving the problem in any module of rank n≥ 2 to solving the problem in any free module of the same rank n. As an application, we show that this can be used to dequantize the LLL algorithm for module lattices presented by Lee et al. (Asiacrypt 2019).