Factoring integers is one of the most analysed problems in number theory and cryptology . Elliptic Curve Method ( ECM ) , which is known by Lenstra method for factoring integers , is one of the best exsitance method for factorization problem . This method introduced in 1987 by Hendrik Willem Lenstra , Jr [45] . For this method , several forms of elliptic curves have been studied , which can be Weierstras curves , Suyama curves , Montgomery curves , Edwards curves and extension families of Edwards curves . The end case is one of the newest of elliptic curves which is known . Using the historical results of Euler and Gauss , Edwards introduced a normal form for elliptic curves and stated the group law in [32] . These curves are defined by the equation . Edwards curves , since to have low cost for group law and memory arithmetic in cryptographic applications , are drawing most attention of cryptologists to their . In recent years , there has been a rapid development of Edwards curves and their application in elliptic curve cryptology . In this article , studied family of Edwards curves and these applications . ECM ability to factoring the "random" integers that interest to number theorists : this method is not as fast as trial division and Pollard's rho method for finding small prime divisers , but it is the method of random choice elliptic curves for finding medium size prime divisers . ECM also ability to factoring the "hard" integers that interest to cryptologists : those integers are factored by sieving methods , which use ECM to find medium size prime divisers of auxiliary integers . ECM ability to find "large" prime divisers; the best record is in 2013 that finds a number 274-bit of the number 947-bit . More data for all recordes of elliptic curve factorization method can be found in \\url{https://www.loria.fr/~zimmerma/records/ecmnet.html}. Many analysis and studies have been done to develop this method , particularly use of the -torsion points and subgroups of elliptic curves which use for ECM . In the final chapter of this article , we stated two steps approach of ECM . Also , the parametrizations been stated that have been introduced well as by Montgomery and Atkin and Morain . These parametrizations provide Weierstras elliptic curves that have -torsion groups and uniformity with and . . Bernstein and Lange have changed these parametrizations to form of Edwards curves .