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 xx and yy coordinates in the generalised curve equation of y2=x3+ax+by^2 = x^3 + ax + b.

@dataclass
class Point:
x: int
y: 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