[Resource Topic] 2025/994: A Fast Multiplication Algorithm and RLWE-PLWE Equivalence for the Maximal Real Subfield of the $2^r p^s$-th Cyclotomic Field

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Title:
A Fast Multiplication Algorithm and RLWE-PLWE Equivalence for the Maximal Real Subfield of the 2^r p^s-th Cyclotomic Field

Authors: Wilmar Bolaños, Antti Haavikko, Rodrigo M. Sánchez-Ledesma

Abstract:

This paper proves the RLWE-PLWE equivalence for the maximal real subfields of the cyclotomic fields with conductor n = 2^r p^s, where p is an odd prime, and r \geq 0 and s \geq 1 are integers. In particular, we show that the canonical embedding as a linear transform has a condition number bounded above by a polynomial in n. In addition, we describe a fast multiplication algorithm in the ring of integers of these real subfields. The multiplication algorithm uses the fast Discrete Cosine Transform (DCT) and has computational complexity \mathcal{O}(n \log n). This work extends the results of Ahola et al., where the same claims are proved for a single prime p = 3.

ePrint: https://eprint.iacr.org/2025/994

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