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 = (xP,yP)
and Q = (xQ,yQ) are not negative of each other,
P +
Q = R where
s = (yP - yQ) / (xP -
xQ)
xR = s2 - xP -
xQ and yR = -yP + s(xP -
xR)
Note that s is the slope of the line through P and
Q.
2.2.2 Doubling the point P
When yP is not
0,
2P = R where
s = (3xP2 + a) /
(2yP )
xR = s2 - 2xP and
yR = -yP + s(xP - xR)
Recall
that a is one of the parameters chosen with the elliptic curve and that s is the
tangent on the point P.