A short basis of the Stickelberger ideal of a cyclotomic field

TL;DR

This paper constructs an explicit short basis for the Stickelberger ideal in cyclotomic fields, providing practical bounds on class numbers and applications in cryptography, especially for lattice problems.

Contribution

It introduces a new explicit short basis for the Stickelberger ideal in cyclotomic fields, enhancing computational and cryptographic applications.

Findings

Explicit short basis for the Stickelberger ideal constructed

Provides an upper bound on the relative class number

Applications in cryptanalysis of lattice problems

Abstract

We exhibit an explicit short basis of the Stickelberger ideal of cyclotomic fields of any conductor $m$, i.e., a basis containing only short elements. By definition, an element of $\mathbb{Z}[G_m]$, where $G_m$ denotes the Galois group of the field, is called short whenever it writes as $\sum_{\sigma\in G_m} \varepsilon_{\sigma}\sigma$ with all $\varepsilon_\sigma\in\{0,1\}$. One ingredient for building such a basis consists in picking wisely generators $\alpha_m(b)$ in a large family of short elements. As a direct practical consequence, we deduce from this short basis an explicit upper bound on the relative class number, that is valid for any conductor. This basis also has several concrete applications, in particular for the cryptanalysis of the Shortest Vector Problem on Ideal lattices.