Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time currently cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((logN)5) to prove the primality of N . An asymptotically fast version of it, attributed to J. O. Shallit, is expected to run in time Õ((logN)4). We describe this version in more detail, leading to actual implementations able to handle numbers with several thousands of decimal digits.