Paper 2020/1020

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

Katharina Boudgoust, Corentin Jeudy, Adeline Roux-Langlois, and Weiqiang Wen


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 power-of-two cyclotomic fields. However, several tools hold for more general classes of number fields and may be of independent interest.

Note: Section 4.1 simplified due to update of reference paper Albrecht and Deo from Asiacrypt'2017.

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Publication info
A minor revision of an IACR publication in ASIACRYPT 2020
Lattice-based cryptographymodule learning with errorsclassical hardnessbinary secret
Contact author(s)
katharina boudgoust @ irisa fr
2021-03-16: last of 2 revisions
2020-08-27: received
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      author = {Katharina Boudgoust and Corentin Jeudy and Adeline Roux-Langlois and Weiqiang Wen},
      title = {Towards Classical Hardness of Module-{LWE}: The Linear Rank Case},
      howpublished = {Cryptology ePrint Archive, Paper 2020/1020},
      year = {2020},
      doi = {10.1007/978-3-030-64834-3_10},
      note = {\url{}},
      url = {}
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