  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 cannot be proven rigorously, but heuristic arguments show that it
should run in time O ((log N)^5) to prove the primality of N. An asymptotically
fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4).
The aim of this article is to describe this version in more details, leading to
actual implementations able to handle numbers with several thousands of decimal
digits.
