Computing points on elliptic curves over finite fields,

Research into elliptic curves over finite fields has boomed since their
recent application in public key cryptography. One of many questions that
one can ask about such a curve is, given an equation for the curve, how to
compute one or more points on it. We will show an efficient algorithm for
this problem, with the main feature that it is completely deterministic,
whereas many known algorithms for finite field computations require the 
use of random bits to be efficient. The algorithm makes use of both the 
geometry of the curve and the multiplicative structure of the ground field.