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Recent research in cryptography has led to the construction of several pseudo-random bit generators, programs producing bits as hard to predict as solving a hard problem. In this paper, 1. We present a new pseudo-random bit generator based on elliptic curves. 2. To construct our generator, we also develop two techniques that are of independent interest: (a) an algorithm that computes the order of an element in an arbitrary Abelian group; and (b) a new oracle proof method for demonstrating the simultaneous security of multiple bits of a discrete logarithm in an arbitrary Abelian group. 3. We present a new candidate hard problem for future use in cryptography: the elliptic logarithm problem.