Two Algorithms To Improve The Efficiency Of Elliptic Curve Cryptosystem

We introduce two efficient algorithms for the elliptic curve cryptography. The first algorithm computes uP+vQ of Koblitz curves. Its complexity is n/2 (n is the extension degree) addition of two points. The result is better than other algorithms we have known. The second algorithm compute 2k P directly from P, where P is a random point on an elliptic curve, without computing the intermediate points, which is faster than k repeated doublings. The algorithm can induce the algorithms of [6] and [7] in the specify condition. Moreover we apply the algorithms to Montgomery form of elliptic curve which was never presented before, the complexity of the algorithm is better than the result which was presented by [16].