## Cryptology ePrint Archive: Report 2019/1109

Revisiting Multivariate Ring Learning with Errors and its Applications on Lattice-based Cryptography

Alberto Pedrouzo-Ulloa and Juan Ramón Troncoso-Pastoriza and Nicolas Gama and Mariya Georgieva and Fernando Pérez-González

Abstract: 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 $n = \prod_i n_i$ into a set of RLWE samples with dimension $\max_i{\{ n_i \}}$. This is especially devastating for low-degree cyclotomics (e.g., $\Phi_4(x) = 1 + x^2$). 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.

Category / Keywords: public-key cryptography / Tensor of Number Fields, Lattice Cryptography, Homomorphic Encryption, Ring Learning with Errors, Multivariate Rings, Hardness Assumptions

Original Publication (with minor differences): Mathematics
DOI:
10.3390/math9080858

Date: received 27 Sep 2019, last revised 21 Apr 2021

Contact author: apedrouzo at gts uvigo es, juan troncoso-pastoriza at epfl ch, nicolas at inpher io, mariya at inpher io, fperez at gts uvigo es

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2019/1109

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