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2021 | OriginalPaper | Buchkapitel

On the Hardness of Module-LWE with Binary Secret

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

Erschienen in: Topics in Cryptology – CT-RSA 2021

Verlag: Springer International Publishing

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Abstract

We prove that the Module Learning With Errors (\(\mathrm {M\text {-}LWE}\)) problem with binary secrets and rank d is at least as hard as the standard version of \(\mathrm {M\text {-}LWE}\) with uniform secret and rank k, where the rank increases from k to \(d \ge (k+1)\log _2 q + \omega (\log _2 n)\), and the Gaussian noise from \(\alpha \) to \(\beta = \alpha \cdot \varTheta (n^2\sqrt{d})\), where 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 \(\sqrt{md}\) in the Gaussian noise, where m is the number of given \(\mathrm {M\text {-}LWE}\) 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.

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Titel: On the Hardness of Module-LWE with Binary Secret
verfasst von: Katharina Boudgoust
Corentin Jeudy
Adeline Roux-Langlois
Weiqiang Wen
Verlag: Springer International Publishing
Buch: Topics in Cryptology – CT-RSA 2021
Print ISBN: 978-3-030-75538-6

Electronic ISBN: 978-3-030-75539-3

Copyright-Jahr: 2021
DOI: https://doi.org/10.1007/978-3-030-75539-3_21

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