{
  "id": "math/0405305",
  "version": "v2",
  "published": "2004-05-15T05:03:49.000Z",
  "updated": "2007-01-11T05:06:44.000Z",
  "title": "A CRT algorithm for constructing genus 2 curves over finite fields",
  "authors": [
    "Kirsten Eisentraeger",
    "Kristin Lauter"
  ],
  "comment": "16 pages. to appear in Proceedings of AGCT-10",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We present a new method for constructing genus 2 curves over a finite field with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discrete-log based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-11T05:06:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G15",
      "11G10",
      "11R37",
      "14G50"
    ],
    "keywords": [
      "finite field",
      "constructing genus",
      "crt algorithm",
      "igusa class polynomials modulo",
      "chinese remainder theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......5305E"
    }
  }
}