  We give a new algorithm for constructing Picard curves over a finite field
with a given endomorphism ring. This has important applications in cryptography
since curves of genus 3 allow for smaller key sizes than elliptic curves. For a
sextic CM-field $K$ containing the cube roots of unity, we define and compute
certain class polynomials modulo small primes and then use the Chinese
Remainder Theorem to construct the class polynomials over the rationals. We
also give some examples.
