2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/121528We 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.16 pages. to appear in Proceedings of AGCT-10Number TheoryAlgebraic Geometry11G15; 11G10; 11R37; 14G50text