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ECM and the Elliott-Halberstam conjecture for quadratic fields

Razvan Barbulescu 1
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03485435
Contributor : Razvan Barbulescu Connect in order to contact the contributor
Submitted on : Friday, December 17, 2021 - 11:50:51 AM
Last modification on : Wednesday, May 25, 2022 - 3:47:03 AM
Long-term archiving on: : Friday, March 18, 2022 - 6:53:21 PM

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Razvan Barbulescu. ECM and the Elliott-Halberstam conjecture for quadratic fields. 2021. ⟨hal-03485435v1⟩

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