Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method

The basic operation in elliptic cryptosystems is the computation of a multiple d?P of a point P on the elliptic curve modulo n. We propose a fast and systematic method of reducing the number of operations over elliptic curves. The proposed method is based on pre-computation to generate an adequate addition-subtraction chain for multiplier the d. By increasing the average length of zero runs in a signed binary representation of d, we can speed up the window method. Formulating the time complexity of the proposed method makes clear that the proposed method is faster than other methods. For example, for d with length 512 bits, the proposed method requires 602.6 multiplications on average. Finally, we point out. that each addition/subtraction over the elliptic curve using homogeneous coordinates can be done in 3 multiplications if parallel processing is allowed.