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§1. Chord-Tangent Computational Methods on a Normal Cubic Curve 25

(1.4) Addition of Two Points. Let E be an elliptic curve defined by the equation in
normal form
2 3 2
y + a 1 xy + a 3 y = f (x) = x + a 2 x + a 4 x + a 6 .
In order to add two points P 1 = (x 1 , y 1 ) and P 2 = (x 2 , y 2 ) we first form the line
through P 1 and P 2 or the tangent line at P 1 when P 1 = P 2 . Consider the third point
of intersection denoted by P 1 P 2 = (x 3 , y 3 ),sothat
P 1 + P 2 =−P 1 P 2 .

Case 1. If x 1 = x 2 so that P 1 = P 2 , then the line through P 1 and P 2 has an
equation y = λx + β, where