Elliptic curve

An elliptic curve is a smooth projective curve of genus one with a specified point. That point gives the curve an algebraic group structure.

Algebraic curve: A curve defined by polynomial equations over a field. Elliptic curves are a particular kind of algebraic curve.
Chord-and-tangent law: A geometric rule for adding points on a plane cubic curve. A line through two points determines their sum through its third intersection.
Elliptic-curve cryptography: Public-key cryptography based on arithmetic in groups of elliptic-curve points. Its security relies on the difficulty of recovering a scalar from point multiplication.
Rational curve: An algebraic curve birationally equivalent to the projective line. Unlike an elliptic curve, a rational curve has genus zero.
Projective space: A geometric space whose points are lines through the origin in a vector space. Projective coordinates include points at infinity needed for the complete curve.
Weierstrass equation: A standard equation describing an elliptic curve over a field. This form makes the geometric group law explicit in coordinates.
Elliptic-curve discrete logarithm problem: The problem of finding an integer scalar from two related elliptic-curve points. This computational problem underlies common elliptic-curve cryptographic systems.
Hyperelliptic curve: A curve admitting a degree-two map to the projective line and having genus at least two. Its higher genus changes both its geometry and arithmetic.
Genus: A topological or algebraic invariant measuring a curve's complexity. Genus one distinguishes elliptic curves from rational curves and higher-genus curves.
Elliptic curve addition: The group operation combining two points on an elliptic curve into another point. The curve's specified point serves as the identity for this operation.

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