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2020 | OriginalPaper | Chapter

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

Authors : Katharina Boudgoust, Corentin Jeudy, Adeline Roux-Langlois, Weiqiang Wen

Published in: Advances in Cryptology – ASIACRYPT 2020

Publisher: Springer International Publishing

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Abstract

We prove that the module learning with errors (\(\mathrm {M\text {-}LWE}\)) 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 (\(\mathrm {LWE}\)) problem. First, we show the classical hardness of \(\mathrm {M\text {-}LWE}\) with an exponential-sized modulus. In a second step, we prove the hardness of \(\mathrm {M\text {-}LWE}\) 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.

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previous chapter Practical Exact Proofs from Lattices: New Techniques to Exploit Fully-Splitting Rings

next chapter Lattice-Based E-Cash, Revisited

https://csrc.nist.gov/Projects/Post-Quantum-Cryptography.

The Python code is publicly available on https://github.com/KatinkaBou/Probabilistic-Bounds-On-Singular-Values-Of-Rotation-Matrices.

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Title: Towards Classical Hardness of Module-LWE: The Linear Rank Case
Authors: Katharina Boudgoust
Corentin Jeudy
Adeline Roux-Langlois
Weiqiang Wen
Publisher: Springer International Publishing
Book: Advances in Cryptology – ASIACRYPT 2020
Print ISBN: 978-3-030-64833-6

Electronic ISBN: 978-3-030-64834-3

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-64834-3_10

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