  The complexity of the elliptic curve method of factorization (ECM) is proven
under the celebrated conjecture of existence of smooth numbers in short
intervals. In this work we tackle a different version of ECM which is actually
much more studied and implemented, especially because it allows us to use
ECM-friendly curves. In the case of curves with complex multiplication (CM) we
replace the heuristics by rigorous results conditional to the
Elliott-Halberstam (EH) conjecture. The proven results mirror recent theorems
concerning the number of primes p such thar p -- 1 is smooth. To each CM
elliptic curve we associate a value which measures how ECM-friendly it is. In
the general case we explore consequences of a statement which translated EH in
the case of elliptic curves.
