Elliptic Curves for Efficient Repeated Additions

In spite of the good security of the cryptosystem on an elliptic curve defined over finite field, the cryptosystem on an elliptic curve is slower than that on a finite field. To be practical, we need a better method to improve a speed of the cryptosystem on an elliptic curve defined over a finite field. In 1991, Koblitz suggested to use an anomalous curve over , which is an elliptic curve with Frobenious map whose trace is 1, and reduced a speed of computation of mP. In this paper, we consider an elliptic curve defined over with Frobenious map whose trace is 3 and suggest an efficient algorithm to compute mP. On the proposed elliptic curve, we can compute multiples mP with +1 addition in worst case.