Cyclotomy Primality Proofs and their Certificates

The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007.