Definition
secp256k1 is the elliptic curve that underpins every Bitcoin private key, public key, and signature. Its name encodes its parameters: sec (Standards for Efficient Cryptography), p256 (a 256-bit prime field), k (a Koblitz curve), and 1 (the first such curve in the standard). The curve equation is the deceptively simple y² = x³ + 7, defined over the finite field of integers modulo the prime p = 2²⁵⁶ − 2³² − 977.
The domain parameters
A secp256k1 implementation is fully specified by six values: the field prime p, the two curve coefficients a = 0 and b = 7, the generator point G, the group order n (a 256-bit prime), and the cofactor h = 1. A cofactor of 1 is cryptographically ideal: every point other than infinity has the full prime order n, so there are no small-subgroup attacks to defend against. The space of valid private keys is therefore the integers from 1 to n − 1.
Why Bitcoin chose it
Two properties made secp256k1 attractive. First, its special non-random construction allows unusually efficient computation, often more than 30% faster than comparable curves. Second, unlike the NIST P-curves, its constants (a = 0, b = 7) are obviously not derived from any hidden seed, removing the suspicion that a backdoor was baked into the parameters. For a system meant to hold value without trusted authorities, that transparency matters.
The curve combines a finite field with point multiplication against a fixed generator point to turn a secret scalar into a public key.
In Simple Terms
secp256k1 is the elliptic curve that underpins every Bitcoin private key, public key, and signature. Its name encodes its parameters: sec (Standards for Efficient Cryptography),…