  We present a new probabilistic algorithm to compute modular polynomials
modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic
curves and are useful in many aspects of computational number theory and
cryptography. Our algorithm has the distinguishing feature that it does not
involve the computation of Fourier coefficients of modular forms. We avoid
computing the exponentially large integral coefficients by working directly
modulo a prime and computing isogenies between elliptic curves via Velu's
formulas.
