Roots of unity in orders

TL;DR

This paper presents polynomial-time algorithms for computing roots of unity and solving discrete logarithms in orders, with implications for cryptology and computer algebra.

Contribution

It introduces deterministic algorithms for roots of unity in orders and solves the discrete logarithm problem in these groups efficiently.

Findings

Algorithms run in polynomial time

Discrete logarithm problem is solvable efficiently

Potential applications in cryptology and computer algebra

Abstract

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.