Secp256k1

Secp256k1 is the elliptic curve on which all bitcoin key pairs live, defined by the equation y^2 = x^3 + 7 over a 256-bit prime field. Standardized by the Standards for Efficient Cryptography Group in 2000, it was an unusual choice when Satoshi Nakamoto selected it in 2009, and it has since become one of the most scrutinized curves in the world.

Why it matters

The curve's security is the wall protecting every bitcoin balance: deriving a private key from a public key requires solving the discrete logarithm problem, infeasible for 256-bit keys with any known classical computer. The choice aged well for a second reason. Secp256k1 is a Koblitz curve whose parameters follow from simple efficiency criteria rather than unexplained constants, so when documents released in 2013 raised concerns that a NIST-standardized random-looking curve component might have been influenced by intelligence agencies, bitcoin's curve was largely outside that suspicion.

How it works

A private key is a number below the curve's group order, and the matching public key is that number multiplied by a fixed generator point. The multiplication is fast in one direction and computationally irreversible in the other. Both ECDSA, used since bitcoin's launch, and Schnorr signatures, added with Taproot in 2021, operate on this same curve, so the eventual practical arrival of large quantum computers, which could attack the discrete logarithm directly, is a shared long-term consideration for the entire system.