Elliptic Curves Defined

Cryptography · Axiom Academy

LESSON Elliptic Curves Defined Understanding the mathematical foundation of elliptic curve cryptography An elliptic curve over the real numbers is defined by the Weierstrass equation: where a and b are constants, and the curve consists of all points (x, y) satisfying this equation, plus a special "point at infinity" denoted by O. This cubic equation creates a smooth, symmetric curve that is fundamental to elliptic curve cryptography. 2. The Discriminant: No Singularities Allowed For a curve to be a valid elliptic curve, it must be non-singular , meaning it has no cusps, self-intersections, or isolated points. This is ensured by the discriminant condition: If this condition is violated, the curve develops singularities (points where the curve is not smooth), which breaks the group structure needed for cryptography. 3. The Point at Infinity: Identity Element Every elliptic curve includes a special point called the point at infinity , denoted O. This point serves as the identity element for the group operation on the curve. The point at infinity can be visualized as the "point where all vertical lines meet" - similar to how parallel lines meet at infinity in projective geometry. 4. The Elliptic Curve Group Law Elliptic curves form an abelian (commutative) group under a geometric addition operation. To add two points P and Q: 1. Draw a line through P and Q 2. Find where this line intersects the curve at a third point R 3. Reflect R across the x-axis to get P + Q

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