ECC Notes: Point at infinity
- Conceptualizing the Point at Infinity:
- The point at infinity, ‘O’, is a special point on an elliptic curve which essentially acts as an identity element with respect to the elliptic curve group law. This means that for any point ‘P’ on the curve, P + O = P.
- It’s called the “point at infinity” because, conceptually, it’s located infinitely far away from all the other points on the curve, in every direction. From a projective geometry perspective, it’s the point where parallel lines ‘meet’.
- This concept is not unique to elliptic curves but is rather common in projective geometry and algebraic geometry, which are the mathematical frameworks in which elliptic curves are typically studied.
- Purpose of the Point at Infinity:
- The introduction of the point at infinity allows us to define the addition operation on the elliptic curve in a way that is closed, meaning that the addition of any two points on the curve always results in another point on the curve. This is important for constructing a well-defined group structure on the curve.
- When considering the operation of point doubling (P+P=2P) on an elliptic curve, when the y-coordinate of the point is zero (y1=0), the tangent line at that point is vertical and would intersect the curve at infinity, so the “double” of that point is defined to be the point at infinity, ‘O’.
- Also, ‘O’ acts as an ‘identity element’ for the group of points on an elliptic curve with the defined addition operation. This means that for any point ‘P’ on the curve, P + O = P and O + P = P.
- Creating the Point at Infinity:
- Strictly speaking, the point at infinity is not ‘created’ but rather is an inherent part of the structure of the elliptic curve when viewed in the projective plane. This is a space that is similar to the normal Cartesian plane, but has an additional ‘line at infinity’ where all the parallel lines meet.
- While in the affine coordinates we work with the equation y² = x³ + ax + b, in the projective plane, the equation of the elliptic curve includes a point at infinity. We typically use homogeneous coordinates [X:Y:Z] for points in the projective plane, and the curve equation becomes Y²Z = X³ + aXZ² + bZ³. The point at infinity is then [0:1:0].
- Thus, the point at infinity is not so much generated as it is inherent in the more complete, projective description of the elliptic curve.
In conclusion, the point at infinity ‘O’ plays a vital role in ensuring the well-definedness of the group operation on an elliptic curve, and is especially important for handling cases where the slope of the line is infinite or the ‘double’ of a point is calculated where the y-coordinate is zero.
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