{
  "id": "2212.11724",
  "version": "v1",
  "published": "2022-12-22T14:15:26.000Z",
  "updated": "2022-12-22T14:15:26.000Z",
  "title": "ECm And The Elliott-Halberstam Conjecture For Quadratic Fields",
  "authors": [
    "Razvan Barbulescu",
    "Florent Jouve"
  ],
  "categories": [
    "cs.CR",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-22T14:15:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliott-halberstam conjecture",
      "quadratic fields",
      "elliptic curve method",
      "cm elliptic curve",
      "proven results mirror"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}