Counting Points on Elliptic Curves Over F2m
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In 1985, Schoof  presented a polynomial time algorithm for computing #E(F q ), the number of Fq-rational points on an elliptic curve E defined over the field F q . The algorithm has a running time of 0(log8 q) bit operations, and is rather cumbersome in practice. Buchmann and Muller  combined Schoof’s algorithm with Shanks’ baby-step giant-step algorithm, and were able to compute orders of curves over Fp, where p is a 27-decimal digit prime. The algorithm took 4.5 hours on a SUN-1 SPARC-station.
KeywordsElliptic Curve Elliptic Curf Abelian Variety Quadratic Residue Modular Equation
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