论文标题
ECM和Elliott-Halberstam的猜想
ECM And The Elliott-Halberstam Conjecture For Quadratic Fields
论文作者
论文摘要
椭圆曲线分解方法（ECM）的复杂性在短时间间隔的平滑数字存在下得到了证明。在这项工作中，我们处理了另一个版本的ECM，该版本实际上对研究和实施了，尤其是因为它允许我们使用ECM友好的曲线。在具有复杂乘法（CM）的曲线的情况下，我们通过严格的结果替换了Elliott-Halberstam（EH）猜想的严格结果。该验证的结果反映了有关Primes p的数量P -1的最新定理。对于每个CM椭圆曲线，我们将一个值相关联，该值可以测量其对ECM的友好程度。在一般情况下，我们探讨了在椭圆曲线的情况下翻译EH的陈述的后果。
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.
