Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.26993

Deterministic and Efficient Ideal Arithmetic via Two-Element Representations

Qi Cheng
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given an ideal in a number field, it is desirable in many situations to find two elements that generate the ideal over the ring of the integers of the field. Existing algorithms are either randomized, or impractical at cryptographic sizes. In the paper, we present a deterministic polynomial time algorithm to find the two-element representation of an ideal. For a monic irreducible integral polynomial \( f(x) \), let \( K=\Q[x]/(f) \) be the number field, and \( O_K \) be the integral closure. Our algorithm works when the norm of the input ideal is co-prime to the index \( [O_K:\Z[x]/f] \). In particular, it handles all ideals for monogenic \( f(x) \), a class that includes the cyclotomic polynomials widely used in lattice based cryptography. A key technical ingredient in our result is a generalized version of Dedekind criterion.

Explore related subjects

math.NTcs.SC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Qi Cheng. 2026-06-30. Deterministic and Efficient Ideal Arithmetic via Two-Element Representations. https://arxiv.org/abs/2606.26993

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗