Elliptic curves are of interest in both coding and cryptography. As curves of genus one, they yield codes with a minimum distance one less than allowed by the Singleton bound. In cryptography, the additive group of points on the curve serves as the underlying group for the implementation of protocols such as key exchange and digital signature. In either case it is of importance to determine precisely the number of points on the curve. This number, by the Hasse-Weil theorem, is within 2 sqrt(q) of q+1, where q is the field size. In cryptography, the field size might be on the order of 10**150 or greater and it is a mathematically challenging problem to determine computationally effective algorithms to achieve this point counting. The talk will review some of the recent work on this problem.

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