Computer Security and Cryptography

15.2 THE ELLIPTIC GROUP OVER THE REALS

The curve y² = x³ − x = x(x − 1)(x + 1) enjoys a property characteristic of all elliptic curves – the chord-tangent group law discovered by Carl Gustav Jacob Jacobi (1804–1851) in the nineteenth century.

Proposition 15.1 (Bezout's Theorem):    If P = (x₁, y₁) and Q = (x₂, y₂) are two points on the elliptic curve y² = x³ + ax + b with 4a³ + 27b² ≠ 0 and if the line PQ joining these points is not vertical, then the line PQ will intersect the curve in a third place ϕ(P, Q) = R = (x₃, − y₃) whose coordinates are given by

and

Proof:    Suppose the equation of the line PQ is

There are two cases to consider; if x₁ ≠ x₂, then

Square y and substitute into the equation y² = x³ + ax + b to obtain

so that

which gives

The case x₁ = x₂ is the limiting case ...