ECm And The Elliott-Halberstam Conjecture For Quadratic Fields

The complexity of the elliptic curve method of factorization (ECM) is proven under a strong conjectural form of existence of friable numbers in short intervals. In the present work we use friability to tackle a different version of ECM which is much more studied and implemented, especially because it enables the use of ECM-friendly curves. In the case of curves with complex multiplication (CM) we replace heuristic arguments by rigorous results conditional on the Elliott–Halberstam (EH) conjecture. The proven results mirror recent work concerning the count of primes p such that p − 1 is friable. In the case of non CM curves, we explore consequences of a hypothetical statement that can be seen as an elliptic curve analogue of EH.

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