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Elliptic Curve Group Law


The group of an elliptic curve which has been transformed to the form

 y^2=x^3+ax+b

is the set of K-rational points, including the single point at infinity. The group law (addition) is defined as follows: Take 2 K-rational points P and Q. Now 'draw' a straight line through them and compute the third point of intersection R (also a K-rational point). Then

 P+Q+R=0

gives the identity point at infinity. Now find the inverse of R, which can be done by setting R=(a,b) giving -R=(a,-b).

This remarkable result is only a special case of a more general procedure. Essentially, the reason is that this type of elliptic curve has a single point at infinity which is an inflection point (the line at infinity meets the curve at a single point at infinity, so it must be an intersection of multiplicity three).


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Cite this as:

Weisstein, Eric W. "Elliptic Curve Group Law." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticCurveGroupLaw.html

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