Curves

The curves module provides a registry of pre-defined elliptic curve parameters commonly used in Elliptic Curve Cryptography (ECC). These curves are standardized by SECG/NIST and have undergone extensive review.

Available Curves

Curve	Key Size	Standard	Common Use
`secp192k1`	192-bit	SECG	Koblitz curve
`secp192r1`	192-bit	NIST P-192	Random curve
`secp224k1`	224-bit	SECG	Koblitz curve
`secp224r1`	224-bit	NIST P-224	Random curve
`secp256k1`	256-bit	SECG	Bitcoin, Ethereum
`secp256r1`	256-bit	NIST P-256	TLS, widely deployed
`secp384r1`	384-bit	NIST P-384	High security
`secp521r1`	521-bit	NIST P-521	Highest security

CurveParams Properties

Each curve is represented as a CurveParams dataclass with the following fields:

Property	Description
`p`	Prime defining the finite field
`a`	Coefficient 'a' in y² = x³ + ax + b
`b`	Coefficient 'b' in y² = x³ + ax + b
`n`	Order of the generator point
`h`	Cofactor
`coord`	Coordinate system (`CoordinateSystem.JACOBIAN` by default)

Functions

`get_curve(name: str) -> CurveParams`

Retrieve curve parameters by standard name (case-insensitive).

from ecutils.curves import get_curve

curve = get_curve("secp256k1")
print(curve.p)   # Prime field
print(curve.n)   # Order
print(curve.a)   # Coefficient a
print(curve.b)   # Coefficient b

Raises KeyError if the curve name is not found.

`get_generator(name: str) -> Point`

Retrieve the generator point G for a named curve. The returned Point includes the curve parameters, so it is ready for arithmetic.

from ecutils.curves import get_curve, get_generator

G = get_generator("secp256k1")
print(G)                # Point(x=..., y=...)
print(G.is_on_curve())  # True

# Scalar multiplication
public_key = 12345 * G

Raises KeyError if the curve name is not found.

Exception Handling

from ecutils.curves import get_curve

try:
    curve = get_curve("invalidCurveName")
except KeyError as e:
    print(e)  # "Unknown curve 'invalidCurveName'. Available: ..."

Notes

For secure applications, use standardized and vetted curves.
All curves conform to SEC 2 / NIST standards.
get_curve() returns CurveParams with Jacobian coordinates by default. Use dataclasses.replace(curve, coord=CoordinateSystem.AFFINE) for affine coordinates.
CurveParams automatically validates the discriminant condition (4a³ + 27b² ≠ 0 mod p) at construction time, rejecting singular curves with ValueError.