The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more studied and implemented, especially because it allows us to use ECM-friendly curves. In the case of curves with complex multiplication (CM) we replace the heuristics by rigorous results conditional to the Elliott-Halberstam (EH) conjecture. The proven results mirror recent theorems concerning the number of primes p such thar p -- 1 is smooth. To each CM elliptic curve we associate a value which measures how ECM-friendly it is. In the general case we explore consequences of a statement which translated EH in the case of elliptic curves.
翻译:计算系数的椭圆曲线方法的复杂性在光滑数字短期存在的著名假设下得到了证明。在这项工作中,我们处理的是不同版本的企业内容管理,实际研究和实施得更多,特别是因为它允许我们使用对企业内容管理友好的曲线。对于具有复杂倍增(CM)的曲线,我们用以Elliott-Halberstam(EH)预测为条件的严格结果取代了超常论。事实证明的结果反映了关于此类单价(1)的质数的最新理论。对于每个内容管理椭圆曲线来说,我们把衡量对企业内容管理友好度的价值联系起来。在一般情况下,我们探索了在椭圆曲线的情况下将环境健康转换的语句的后果。