Paper 2024/1899
Fast Multiplication and the PLWE-RLWE Equivalence for an Infinite Family of Cyclotomic Subextensions
Abstract
We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the $2^r 3^s$-th cyclotomic field for $r \geq 3$ and $s \geq 1$. Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with $\mathcal{O}(n \log n)$ arithmetic operations, where $n$ is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the $p$-th cyclotomic field versus the maximal totally real subextension of the $4p$-th cyclotomic field for a reasonable set of parameters of cryptographic size.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Published elsewhere. arXiv
- DOI
- 10.48550/arXiv.2410.00792
- Keywords
- Number Theoretic TransformDiscrete Cosine TransformRing Learning with ErrorsPolynomial Learning with Errors
- Contact author(s)
-
joonas ahola @ aalto fi
ivan blancoc @ uah es
wilmar bolanos @ aalto fi
anttijhaavikko @ gmail com
camilla hollanti @ aalto fi
rodrma01 @ ucm es - History
- 2024-11-25: approved
- 2024-11-22: received
- See all versions
- Short URL
- https://ia.cr/2024/1899
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2024/1899,
author = {Joonas Ahola and Iván Blanco-Chacón and Wilmar Bolaños and Antti Haavikko and Camilla Hollanti and Rodrigo M. Sánchez-Ledesma},
title = {Fast Multiplication and the {PLWE}-{RLWE} Equivalence for an Infinite Family of Cyclotomic Subextensions},
howpublished = {Cryptology {ePrint} Archive, Paper 2024/1899},
year = {2024},
doi = {10.48550/arXiv.2410.00792},
url = {https://eprint.iacr.org/2024/1899}
}