Paper 2022/472

On the Hardness of Module Learning With Errors with Short Distributions

Katharina Boudgoust, Aarhus University
Corentin Jeudy, Orange Labs, Applied Crypto Group, Univ Rennes, CNRS, IRISA
Adeline Roux-Langlois, Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC
Weiqiang Wen, LTCI, Telecom Paris, Institut Polytechnique de Paris

The Module Learning With Errors problem (M-LWE) is a core computational assumption of lattice-based cryptography which offers an interesting trade-off between guaranteed security and concrete efficiency. The problem is parameterized by a secret distribution as well as an error distribution. There is a gap between the choices of those distributions for theoretical hardness results (standard formulation of M-LWE, i.e., uniform secret modulo $q$ and Gaussian error) and practical schemes (small bounded secret and error). In this work, we make progress towards narrowing this gap. More precisely, we prove that M-LWE with uniform $\eta$-bounded secret for any $1 \leq \eta \ll q$ and Gaussian error, in both its search and decision variants, is at least as hard as the standard formulation of M-LWE, provided that the module rank $d$ is at least logarithmic in the ring degree $n$. We also prove that the search version of M-LWE with large uniform secret and uniform $\eta$-bounded error is at least as hard as the standard M-LWE problem, if the number of samples $m$ is close to the module rank $d$ and with further restrictions on $\eta$. The latter result can be extended to provide the hardness of M-LWE with uniform $\eta$-bounded secret and error under specific parameter conditions. Overall, the results apply to all cyclotomic fields, but most of the intermediate results are proven in more general number fields.

Note: This paper contains novel results and generalizations of existing ones already published in Boudgoust et al. (Asiacrypt'20) and Boudgoust et al. (CT-RSA'21)

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Published by the IACR in JOC 2022
Lattice-Based Cryptography Module Learning With Errors Short Distributions Bounded Secret Bounded Error
Contact author(s)
katharina boudgoust @ cs au dk
corentin jeudy @ irisa fr
adeline roux-langlois @ cnrs fr
weiqiang wen @ telecom-paris fr
2022-12-01: revised
2022-04-22: received
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      author = {Katharina Boudgoust and Corentin Jeudy and Adeline Roux-Langlois and Weiqiang Wen},
      title = {On the Hardness of Module Learning With Errors with Short Distributions},
      howpublished = {Cryptology ePrint Archive, Paper 2022/472},
      year = {2022},
      doi = {10.1007/s00145-022-09441-3},
      note = {\url{}},
      url = {}
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