In this report we study the problem of counting the number of points on an elliptic curve over a finite prime field. This problem is not only very interesting for number theorists but has recently gained a lot of attention among cryptographers. Recently, there has been a lot of progress concerning the problem of computing this group order #E(Fp). The algorithm of Atkin-Elkies has been partially improved and implemented in Paris and Saarbrücken. In both implementations the algorithm is distributed over a network of workstations by means of the system sc LiPS which supports such distributions. The current record is the computation of the group order #E(Fp), where p is a 375-digit prime. That computation took approximately 1765 MIPS days. In this report we briefly describe the state of the art of counting points on elliptic curve over finite prime fields. We explain the main computational problems and their solution by means of distributed and parallel computation
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