Cyclotomy Primality Proving - Recent Developments

Primality proving by cyclotomy is an extension of the Jacobi sum primality test, initially proposed by Adleman, Rumely and Pomerance [3] and implemented by H. Cohen and A. Lenstra [7]. In his presentation of the algorithm of Adleman, Rumely and Pomerance at the Bourbaki Seminar 1981 [14], H. W. Lenstra Jr. proposed under the name of “Galois theory test” the idea to combine classical Lucas — Lehmer tests with the Jacobi sum test. This idea was first studied and implemented by Bosma and van der Hulst in their thesis [6]. In our recently completed thesis [19], we considered the topic anew, from a slightly changed perspective and made an implementation which allowed establishing new general primality testing records. In this paper we shall give an overview of cyclotomy from the perspective of the recent research and implementation. We also discuss the drawbacks of the algorithm — the overpolynomial run time and lack of certificates — and mention some open problems which may lead to future improvements.