Parallel Algorithm for Multiplication on Elliptic Curves

Given a positive integer n and a point P on an elliptic curve E, the computation of nP , that is, the result of adding n times the point P to itself, called the scalar multiplication, is the central operation of elliptic curve cryptosystems. We present an algorithm that, using p processors, can compute nP in time O(log n+H(n)=p+ log p), where H(n) is the Hamming weight of n. Furthermore, if this algorithm is applied to Koblitz curves, the running time can be reduced to O(H(n)=p+ log p).