Generator Point

The generator point, written G, is the fixed, publicly known starting point on the secp256k1 elliptic curve from which every Bitcoin key pair is derived. A private key is simply a number k between 1 and the curve's order n, a value slightly below 2 to the power 256, and the corresponding public key is the point k times G, computed by repeated point addition on the curve.

Why it matters

All of Bitcoin's ownership reduces to one asymmetry anchored at G. Multiplying the generator point by a private key is fast, taking microseconds, but reversing the operation, recovering k from the public key k times G, is the elliptic curve discrete logarithm problem, computationally infeasible with any known technology at this key size. Every signature that spends bitcoin is a proof of knowing k without revealing it. Because every user on earth derives keys from the same G, the parameter's provenance matters: secp256k1's constants are notably plain, chosen efficiently rather than from unexplained seed values, a transparency that spared it the suspicion that fell on some NIST curves after 2013.

How it works

The curve is defined by the equation y squared equals x cubed plus 7 over a 256 bit prime field. Adding G to itself repeatedly walks through a cyclic group of nearly 2 to the power 256 points, and efficient multiplication uses doubling and adding, requiring only about 256 doublings for any key. The same operation executes identically in every wallet, from hardware devices to Bitcoin Core.