Roots of unity in orders
This addresses computational problems in algebraic structures with potential applications in cryptology and computer algebra.
The paper presents deterministic polynomial-time algorithms for computing primitive idempotents and generators for the group of roots of unity in orders, and shows that the discrete logarithm problem in this group can be solved in polynomial time.
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.