Parametrizations for Families of ECM-Friendly Curves

Abstract

Abstract. We provide a new family of elliptic curves that results in a one to two percent performance improvement of the elliptic curve integer factorization method. The speedup is confirmed by extensive tests for factors ranging from 15 to 63 bits.

Modular curves over number fields and ECM

“…As a result, a criterion for computing the friendliness of the reduction of E/Q was developed. High values of it were found, going further than simply having large torsion group [28]. The work was extended to the reduction of a rational curve E over some quadratic field in [8].…”

A classification of ECM-friendly families of elliptic curves using modular curves

“…(see row 5 and row 18 in Table 4). more Also, Gélin et al [GKL17] proposed subfamilies with points of infinite order on the families of [BBB + 13]. We do not discuss further the case of subfamilies with points of infinite order because they are not known to modify the behaviour of ECM and because they produce curves of large coefficients.…”

“…As discussed in Appendix A, an important application of ECM consists in using the same elliptic curve to test smoothness of many integers. In this context, several articles [BBL10], [HMR16], [GKL17], [Mon87] measure the quality of a curve E for the ECM algorithm as the proportion of…”

Substantiation of correctness and advantages of Lenstra factorization method on Edwards curves

“…The Lenstra method [4][5][6][7][8] is one of the fastest general methods of integer number factorization. It can be used both as a separate factorization method, and as one of the stages of the "sifting" methods, which are the fastest factorization methods [9][10][11]. This algorithm uses the elliptic curve over the rational numbers field and some its reduction by the integer modulo, and requires about 2log 2 k additions of the points of this curve, where k is the algorithm parameter, the choice of which defones the probability of success of the algorithm.…”

“…Bernstein's ideas received further development in papers [2,10,11]. In 2013, using the Lenstra method, it became possible to find 274-bit divisor for 787-bit number (7 337 +1).…”