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 $x$ and $y$ coordinates in the generalised curve equation of $y^2 = x^3 + ax + b$ .

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).

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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