JKL-ECM: an implementation of ECM using Hessian curves

We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion Z/6Z⊕Z/6Z and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over Q and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion Z/2Z⊕Z/8Z of Bernstein et al. are completely recovered by the Z/4Z ⊕ Z/8Z family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.