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.

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

P + Q = R where 

s = (y_P - y_Q) / (x_P - x_Q) 

x_R = s² - x_P - x_Q and y_R = -y_P + s(x_P - x_R) 

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

When y_P is not 0, 

2P = R where 

s = (3x_P² + a) / (2y_P ) 

x_R = s² - 2x_P and y_R = -y_P + s(x_P - x_R) 

Recall that a is one of the parameters chosen with the elliptic curve and that s is the tangent on the point P. 

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