Elliptic Curve in Python

Recall that an elliptic curve over a finite field has 3 distinct properties — aa, bb, and the field parameters. Let's define them below:

@dataclass
class EllipticCurve:
a: int
b: int
field: PrimeGaloisField
def __contains__(self, point: "Point") -> bool:
x, y = point.x, point.y
return y ** 2 == x ** 3 + self.a * x + self.b
def __post_init__(self):
# Encapsulate int parameters in FieldElement
self.a = FieldElement(self.a, self.field)
self.b = FieldElement(self.b, self.field)
# Check for membership of curve parameters in the field.
if self.a not in self.field or self.b not in self.field:
raise ValueError

Defining secp256k1

secp256k1 refers to the parameters of the elliptic curve used in Bitcoin's public-key cryptography. The name represents the specific parameters of curve:

  • sec: stands for Standards for Efficient Cryptography.

  • p: indicates that what follows are the parameters of the curve.

  • 256: length in bits of the field size.

  • k: Kolbitz curve, as opposed to random. The non-random construction allows for efficient construction.

  • 1: sequence number

The general equation of the secp256k1 curve is y2=x3+7y^2 = x^3 + 7. Let's represent this curve using the classes we have already defined.

# Ref: https://en.bitcoin.it/wiki/Secp256k1
# secp256k1 elliptic curve equation: y² = x³ + 7
# Prime of the finite field
P: int = (
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
)
field = PrimeGaloisField(prime=P)
# Elliptic curve parameters A and B of the curve : y² = x³ Ax + B
A: int = 0
B: int = 7
secp256k1 = EllipticCurve(
a=A,
b=B,
field=field
)