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where k is the private key, G is the generator point, and K is the resulting public key, a point on the curve. Since 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 rela‐ tionship 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). 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 above, we take the private key generated previously and multiply it by G: Multiply the private key k with the generator point G to find the public key K. K = 1E99423A4ED27608A15A2616A2B0E9E52CED330AC530EDCC32C8FFC6A526AEDD * G Public Key K defined as a point K = (x,y). K = (x, y) where, x = F028892BAD...DC341A y = 07CF33DA18...505BDB To visualize multiplication of a point with an integer, we will use the simpler elliptic curve over the real numbers — remember, the math is the same. Our goal is to find the multiple kG of the generator point G. That 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. The diagram below shows the process for deriving G, 2G, 4G, as a geometric operation on the curve. Introduction | 69PDF Image | Mastering Bitcoin
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