for some . This exponent may be decreased to for some versions by heuristic arguments. ECPP works the same way as most other primality tests do, finding a group and showing its size is such that is prime. For ECPP the group is an elliptic curve over a finite set of quadratic forms such that is trivial to factor over the group.
ECPP generates an Atkin-Goldwasser-Kilian-Morain certificate of primality by recursion and then attempts to verify the certificate. The step that takes the most CPU time is the certificate generation, because factoring over a class field must be performed. The certificate can be verified quickly, allowing a check of operation to take very little time.
As of 2011 the largest prime  that has been proved with ECPP method is the 26,643-digits prime value of the Ramanujan tau function: 
The distributed computation with fastECPP software by François Morain started in January 2011 and ended in April 2011. The total CPU time is equal to 2355 days.