Keys and Bitcoin Addresses

One of the ways that Bitcoin uses cryptographic keys is in generating Bitcoin addresses, which are often derived from public keys. This chapter covers public keys and how they are used to generate addresses.

Generating a Public Key

Starting with a private key in the form of a randomly generated number $k$ , we multiply it by a predetermined point on the curve called the generator point $G$ to produce another point somewhere else on the curve, which is the corresponding public key $K$ . The generator point is specified as part of the secp256k1 standard and is always the same for all keys in bitcoin:

$\begin{equation} {K = k * G} \end{equation}$

where $k$ is the private key, $G$ is the generator point, and $K$ is the resulting public key, a point on the curve. Because the generator point is always the same for all bitcoin users, a private key $k$ multiplied with $G$ will always result in the same public key $K$ . The relationship between k and K is fixed, but can only be calculated in one direction, from $k$ to $K$ . That's why a bitcoin address (derived from $K$ ) can be shared with anyone and does not reveal the user's private key ( $k$ ).

Tip: A private key can be converted into a public key, but a public key cannot be converted back into a private key because the math only works one way.

Implementing the elliptic curve multiplication, we take the private key $k$ generated previously and multiply it with the generator point $G$ to find the public key $K$ :

K = 1E99423A4ED27608A15A2616A2B0E9E52CED330AC530EDCC32C8FFC6A526AEDD * G

Public key $K$ is defined as a point $\text{K = (x,y)}$ :

K = (x, y)

where,

x = F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A
y = 07CF33DA18BD734C600B96A72BBC4749D5141C90EC8AC328AE52DDFE2E505BDB

To visualize multiplication of a point with an integer, we will use the simpler elliptic curve over real numbers - remember, the math is the same. Our goal is to find the multiple $kG$ of the generator point $G$ , which is the same as adding $G$ to itself, $k$ times in a row. In elliptic curves, adding a point to itself is the equivalent of drawing a tangent line on the point and finding where it intersects the curve again, then reflecting that point on the x-axis.

Elliptic curve cryptography: visualizing the multiplication of a point $G$ by an integer $k$ on an elliptic curve shows the process for deriving $G$ , $2G$ , $4G$ , and $8G$ as a geometric operation on the curve.

Tip: Bitcoin uses the secp256k1 optimized C library to do the elliptic curve math.

Figure 4. Elliptic curve cryptography: visualizing the multiplication of a point $G$ by an integer $k$ on an elliptic curve