We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and the
ir Galois closures, we pursue an approach due to Dupont for evaluating $theta$- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example with class number 17608.
The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be mo
delled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties.
