Starfish on strike

Abstract This paper improves the price-performance ratio of ECM, the elliptic-curve method of integer factorization. In particular, this paper constructs a¿=¿-¿1 twisted Edwards curves having Q-torsion group Z/2 x Z/4, Z/8, or Z/6 and having a known non-torsion point; demonstrates that, compared to the curves used in previous ECM implementations, some of the new curves are more effective at finding small primes despite being faster; and precomputes particularly effective curves for several specific sizes of primes. Keywords: Factorization - ECM - elliptic-curve method - curve selection - Edwards coordinates - twisted Edwards curves - Suyama curves