arXiv subjects

Antti Haavikko

Publications and source records attributed to Antti Haavikko.

3 recordsLinked to original sources

arXiv2026

On the computation of base-change lifts and lifts of Hida families

We derive an explicit formula for the Hecke eigenvalues of a Hilbert modular form which is a base-change lift of a classical newform to a totally real number field. We show that for a totally real Galois number field $F$ the $L$-function of a base-change lifted form can be factorized as a product of twists of the $L$-function of the underlying classical form over irreducible representations of $\mathrm{Gal}(F / \mathbb{Q})$. Moreover, we use the formula for the Hecke eigenvalues of a base-change lift to prove the existence of a base-change lift of a Hida family. In particular, we show that a Hida family of classical Hecke eigenforms can be lifted to a formal power series that specializes to the base-change lifts of the Hida family of classical cusp forms.

math.NT↗

arXiv2025

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

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 $\mathcal{O}(n \log n)$. Both the proof of the RLWE-PLWE equivalence and the fast multiplication algorithm are generalizations of previous results by Ahola et al., where the same claims are proved for a single prime $p = 3$.

cs.CR↗

arXiv2025

Fast Multiplication and the PLWE-RLWE Equivalence for an Infinite Family of Maximal Real Subfields of Cyclotomic Fields

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.

cs.CR↗