Double-and-add

Reduction of scalar multiplication into double and add operations

Motivation

Additions (including doubles) are cheap for elliptic curve points. As such, we try to reduce scalar multiplication (or elliptic curve points multiplied by a scalar) to a series of additions.

Algorithm Process

In practice, Double-and-add is implemented as follows:

// Scalar multiplication: Double-and-Add
Point ScalarMultiply(Point p, int k) {
    Point result; // point at infinity
    Point powerOfP = p;
    if (k & 1) {             
        result = result + powerOfP; 
    }
    k >>= 1; 
    while (k > 0) {
        powerOfP = powerOfP.Double(); 
        if (k & 1) {             
            result = result + powerOfP; 
        }
        k >>= 1; 
    }
    return result;
}

For example, let's take a look at 6:

$$6 = \underbrace{0}_{2^0}\underbrace{1}_{2^1}\underbrace{1}_{2^2}$$

k = 6
result = 0;

// 1st while loop:
powerOfP = powerOfP.Double()  // 1 double : powerOfP = 2p
result += powerOfP            // 1 add    : result = 2p

// 2nd while loop:
powerOfP = powerOfP.Double()  // 1 double : powerOfP = 4p
result += powerOfP            // 1 add    : result = 6p

= 2 adds + 2 doubles

Written by Ashley Jeong of Fractalyze