Computing the public key

Now that we have defined point doubling, we can calculate a point on the curve that is the multiple of the given point generator, point G (for example, 4G = G + G + G + G), and this could be computed using point doubling.

Let's use this concept to compute a public-private key pair in an asymmetric cryptography system.

Every curve domain parameter is the same for a given specification. Refer to the technical specifications of the secp256k1 standard in a later section of this chapter that is used in Bitcoin and other blockchain applications' digital signature algorithms. Let's say k is a randomly chosen private key, and K is the public key to be generated. The generator of the curve, G, has a standard value. The public key could be computed by performing the following operation on the curve:

K = k*G

We can generate the public key using this equation on an elliptic curve using point doubling. Point doubling on an elliptic curve is a one-way operation. It is, therefore, a challenging task to compute the multiplied value k after the required point K has been found:

Figure 2.9 shows the process of multiplying an integer value by the base point G. In this case, points 2G and 4G are derived using point doubling of G on the given curve. This geometrical method could be used to generate the public key, K, by multiplying the generator by the private key k times.