Towards Classical Hardness of Module-LWE: The Linear Rank Case

Katharina Boudgoust (1) , Corentin Jeudy (1) , Adeline Roux-Langlois (1) , Weiqiang Wen (1)

1 EMSEC - Embedded Security and Cryptography / Sécurité cryptographie embarquée

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.

Mots clés: Lattice-based cryptography; module learning with errors; classical hardness; binary secret

Domaines: Cryptographie et sécurité [cs.CR]

Katharina Boudgoust, Corentin Jeudy, Adeline Roux-Langlois, Weiqiang Wen. Towards Classical Hardness of Module-LWE: The Linear Rank Case. Asiacrypt 2020 - 26th Annual International Conference on the Theory and Application of Cryptology and Information Security, Dec 2020, Virtual, France. ⟨hal-03038053⟩