Elliptic Curve Addition: An Algebraic Approach

2.2 Elliptic Curve Addition: An Algebraic Approach

Although the previous geometric descriptions of elliptic curves provides an excellent method of illustrating elliptic curve arithmetic, it is not a practical way to implement arithmetic computations. Algebraic formulae are constructed to efficiently compute the geometric arithmetic.

2.2.1 Adding distinct points P and Q

When P = (x_P,y_P) and Q = (x_Q,y_Q) are not negative of each other,

P + Q = R where

l = (y_P - y_Q) / (x_P - x_Q)

x_R = l² - x_P - x_Q
and
y_R = -y_P + l(x_P - x_R)

Note that l is the slope of the line through P and Q.

2.2.2 Doubling the point P

When y_P is not 0,

2P = R where

l = (3x_P² + a) / (2y_P )

x_R = l² - 2x_P
and
y_R = -y_P + l(x_P - x_R)

Recall that a is one of the parameters chosen with the elliptic curve and that l is the slope of the line through P and Q.

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