Prime Numbers

344 Chapter 7 ELLIPTIC CURVE ARITHMETIC
Algorithm 7.4.4 (Inversionless ECM). Given a composite number n to be
factored, with gcd(n, 6) = 1, this algorithm attempts to uncover a nontrivial
factor of n. This algorithm is inversion-free, needing only large-integer multiplymod
(but see text following).
1. [Choose criteria]
B1 = 10000; // Stage-one limit (must be even).
B2 = 100B1; // Stage-two limit (must be even).
D = 100; // Total memory is about 3D size-n integers.
2. [Choose random curve Eσ]
Choose random σ ∈ [6,n− 1]; // Via Theorem 7.4.3.
u =(σ 2 − 5) mod n;
v =4σ mod n;
C =((v − u) 3 (3u + v)/(4u 3 v) − 2) mod n;
// Note: C determines curve y 2 = x 3 + Cx 2 + x,
// yet, C can be kept in the form num/den.
Q =[u 3 mod n : v 3 mod n]; // Initial point is represented [X : Z].
3. [Perform stage one]
for(1 ≤ i ≤ π(B1)) { // Loop over primes pi.
Find largest integer a such that pa i ≤ B1;
Q =[pa i ]Q; // Via Algorithm 7.2.7, and perhaps use FFT
enhancements (see text following).
}
g =gcd(Z(Q),n); // Point has form Q =[X(Q) :Z(Q)].
if(1