4.1 The right-to-left binary method

Suppose E is an elliptic curve defined over a prime field Fp. The equation of E is given by E: y <sup>2</sup> <sup>=</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> ax <sup>þ</sup> <sup>b</sup> (mod <sup>p</sup>). Let <sup>P</sup> <sup>=</sup> (x, <sup>y</sup>) be <sup>a</sup> generator point that lies on E which has a (large) prime order n. Choosing v to compute vP can be done from the range [1, n�1]. So, it should first write v in a binary representation string (et-1, …, e1, e0)2. The starting will be happened with a point Q in E (Fp), (that is, Q = ∞). With the i index that takes the values 0, 1, …, t � 1, the computation of Q = Q þ P can be done if ei = 1. After then, the value 2P is computed and plugging 2P by P. The processing continues until the last value t � 1. Therefore, the last computed value of a point Q is the scalar multiplication point vP [1]. The summary of the RLB method can be given in the following algorithm.
