COMPUTING GENUS-2 CURVES USING GENERAL ISOGENIES

摘要

An Igusa class polynomial over rational numbers is computed from a set of Igusa class polynomials modulo a set of small primes. The set of Igusa class polynomials modulo a set of small primes is computed by finding all of the maximal curves in the isogeny class for each of the small primes. In particular, for each prime number in a set of prime numbers, a curve in the isogeny class for the prime number is identified, for example through a random search. Given a curve in this isogeny class, isogenies of general degree are applied to the identified curve, until an initial maximal curve, i.e., a curve with a maximal endomorphism ring, is found in this isogeny class. After an initial maximal curve in the isogeny class is found, all other maximal curves in this isogeny class are found by applying isogenies of general degree to the initial maximal curve. This set of maximal curves for the set of prime numbers defines set of Igusa class polynomials modulo the small primes. A Chinese remainder approach is then applied to construct an Igusa class polynomial over the rational numbers from the computed set of Igusa class polynomials modulo the small primes.