elliptic curve
A curve of the form `y² = x³ + ax + b` (with `4a³ + 27b² ≠ 0`, so the curve has no cusps or self-intersections). When taken over a finite field `F_p`, the affine solutions plus a single _point at infinity_ form a finite abelian group under the chord-and-tangent construction inherited from Bezout's theorem. ECDSA, EdDSA, and most modern public-key cryptography stand on this group. Bitcoin uses the specific curve `y² = x³ + 7` over the prime `p = 2²⁵⁶ − 2³² − 977`, called secp256k1.
1830s pure math · 1985 cryptography · Niels Abel & Carl Jacobi (1830s); Neal Koblitz & Victor Miller (1985) · Norway / Königsberg, then UC / IBM
Abel and Jacobi studied elliptic curves in the 1830s as inverse functions of elliptic integrals — pure mathematics with no application in sight. 150 years later, in 1985, Koblitz and Miller independently realized the chord-and-tangent group law on these curves gave a cryptographic one-way function. Bitcoin's signature scheme is a direct descendant.