  We present a very efficient algorithm to construct an elliptic curve E and a
finite field F such that the order of the point group E(F) is a given prime
number N. Heuristically, this algorithm only takes polynomial time Otilde((\log
N)^3), and it is so fast that it may profitably be used to tackle the related
problem of finding elliptic curves with point groups of prime order of
prescribed size. We also discuss the impact of the use of high level modular
functions to reduce the run time by large constant factors and show that recent
gonality bounds for modular curves imply limits on the time reduction that can
be obtained.
