The Multi-Primes Lanstra's Elliptic Curve Factorization Algorithm

The Lenstra's elliptic curve factorization (LEF) algorithm is a problem to factor a composite number N that is a modulus to define an elliptic curve over it, namely E mod N. The LEF method considered as the third fastest algorithm comparing to multiple polynomial quadratic and general number field sieves algorithms. On the LEF, the number N factors into two primes p and q based on the elliptic curve arithmetic. This arithmetic consists of two operations, doubling and addition points on E mod N. Using these operations, the computations 2 P, 3P, ..., mp = ∞, where p is a point on E, are done. Two curves E mod p, and E mod q are determined. On one of these curves an efficient calculating is occurred. In this work, the previous version of the LEF algorithm can be extended to factor N into multi-primes p, q, r, ..., n. The points 2P, 3P, ..., mP=∞ can be computed on n curves, namely E mod p, E mod q, E mod r, ..., E mod n to determine which curve is an efficient for computing. Any algorithm for factoring N through applying the proposed multi-primes Lenstra's elliptic curve factorization (M-PLEF) algorithm is much more difficult. So, the M-PLEF algorithm is more secure in compare to LEF algorithm.