BARBULESCU, Razvan; JOUVE, Florent

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BARBULESCU, Razvan
Lithe and fast algorithmic number theory [LFANT]
Analyse cryptographique et arithmétique [CANARI]

JOUVE, Florent
Institut de Mathématiques de Bordeaux [IMB]

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Acta Arithmetica. 2024

Instytut Matematyczny PAN

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2024

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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.< Réduire

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Elliott-Halberstam conjecture

Elliptic curve method

Number field sieve

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ECM And The Elliott-Halberstam Conjecture For Quadratic Fields

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