We prove that the module learning with errors (M-LWE) problem with arbitrary polynomial-sized modulus p is classically at least as hard as standard worst-case lattice problems, as long as the module rank d is not smaller than the number field degree n. Previous publications only showed the hardness under quantum reductions. We achieve this result in an analogous manner as in the case of the learning with errors (LWE) problem. First, we show the classical hardness of M-LWE with an exponential-sized modulus. In a second step, we prove the hardness of M-LWE using a binary secret. And finally, we provide a modulus reduction technique. The complete result applies to the class of powerof-two cyclotomic fields. However, several tools hold for more general classes of number fields and may be of independent interest.

We prove that the Module Learning With Errors (M-LWE) problem with binary secrets and rank d is at least as hard as the standard version of M-LWE with uniform secret and rank k, where the rank increases from k to d ≥ (k + 1) log 2 q + ω(log 2 n), and the Gaussian noisewhere n is the ring degree and q the modulus. Our work improves on the recent work by Boudgoust et al. in 2020 by a factor of √ md in the Gaussian noise, where m is the number of given M-LWE samples, when q fulfills some number-theoretic requirements. We use a different approach than Boudgoust et al. to achieve this hardness result by adapting the previous work from Brakerski et al. in 2013 for the Learning With Errors problem to the module setting. The proof applies to cyclotomic fields, but most results hold for a larger class of number fields, and may be of independent interest.

No abstract

The Module Learning With Errors problem (M-LWE) has gained popularity in recent years for its security-efficiency balance, and its hardness has been established for a number of variants. In this paper, we focus on proving the hardness of (search) M-LWE for general secret distributions, provided they carry sufficient min-entropy. This is called entropic hardness of M-LWE. First, we adapt the line of proof of Brakerski and Döttling on R-LWE (TCC'20) to prove that the existence of certain distributions implies the entropic hardness of M-LWE. Then, we provide one such distribution whose required properties rely on the hardness of the decisional Module-NTRU problem.

No abstract