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Representing a point

We call a Point an element of the elliptic curve group, which we can represent with its coordinates. Let's define a Point class to encapsulate the xxx and yyy coordinates in the generalised curve equation of y2=x3+ax+by^2 = x^3 + ax + by2=x3+ax+b.
from typing import Optional
​
@dataclass
class Point:
    x: Optional[int]
    y: Optional[int]
​
    curve: EllipticCurve
​
    def __post_init__(self):
        # Ignore validation for I
        if self.x is None and self.y is None:
            return
​
        # Encapsulate int coordinates in FieldElement
        self.x = FieldElement(self.x, self.curve.field)
        self.y = FieldElement(self.y, self.curve.field)
​
        # Verify if the point satisfies the curve equation
        if self not in self.curve:
            raise ValueError
Point-compression: To reduce the storage size for a curve point, one can also store a sign and thex coordinate. It is then possible to  reconstruct the y by calculating sign * sqrt(x^3+a*x+b).
Ref: https://bitcoin.stackexchange.com/a/29905​
Elliptic Curve in Python
Group Theory
Last modified 2mo ago