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Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography

1
AtlanTTic Research Center, Universidade de Vigo, 36310 Vigo, Spain
2
Laboratory for Data Security, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
3
Inpher, CH-1015 Lausanne, Switzerland
*
Author to whom correspondence should be addressed.
Academic Editor: Askar Tuganbaev
Mathematics 2021, 9(8), 858; https://doi.org/10.3390/math9080858
Received: 28 February 2021 / Revised: 8 April 2021 / Accepted: 9 April 2021 / Published: 14 April 2021
(This article belongs to the Section Mathematics and Computer Science)
The “Multivariate Ring Learning with Errors” problem was presented as a generalization of Ring Learning with Errors (RLWE), introducing efficiency improvements with respect to the RLWE counterpart thanks to its multivariate structure. Nevertheless, the recent attack presented by Bootland, Castryck and Vercauteren has some important consequences on the security of the multivariate RLWE problem with “non-coprime” cyclotomics; this attack transforms instances of m-RLWE with power-of-two cyclotomic polynomials of degree n=ini into a set of RLWE samples with dimension maxi{ni}. This is especially devastating for low-degree cyclotomics (e.g., Φ4(x)=1+x2). In this work, we revisit the security of multivariate RLWE and propose new alternative instantiations of the problem that avoid the attack while still preserving the advantages of the multivariate structure, especially when using low-degree polynomials. Additionally, we show how to parameterize these instances in a secure and practical way, therefore enabling constructions and strategies based on m-RLWE that bring notable space and time efficiency improvements over current RLWE-based constructions. View Full-Text
Keywords: tensor of number fields; lattice cryptography; homomorphic encryption; ring learning with errors; multivariate rings tensor of number fields; lattice cryptography; homomorphic encryption; ring learning with errors; multivariate rings
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MDPI and ACS Style

Pedrouzo-Ulloa, A.; Troncoso-Pastoriza, J.R.; Gama, N.; Georgieva, M.; Pérez-González, F. Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography. Mathematics 2021, 9, 858. https://doi.org/10.3390/math9080858

AMA Style

Pedrouzo-Ulloa A, Troncoso-Pastoriza JR, Gama N, Georgieva M, Pérez-González F. Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography. Mathematics. 2021; 9(8):858. https://doi.org/10.3390/math9080858

Chicago/Turabian Style

Pedrouzo-Ulloa, Alberto, Juan R. Troncoso-Pastoriza, Nicolas Gama, Mariya Georgieva, and Fernando Pérez-González. 2021. "Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography" Mathematics 9, no. 8: 858. https://doi.org/10.3390/math9080858

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