BLS12-381 Curve

# Generator points
G1  # Generator point on G1 (base curve)
G2  # Generator point on G2 (twist curve)  
G12 # Identity element in GT (target group)

# Points at infinity
Z1  # Point at infinity for G1
Z2  # Point at infinity for G2

# Curve parameters
b: int      # Curve coefficient for G1: y^2 = x^3 + b
b2          # Curve coefficient for G2 (quadratic extension)
b12         # Curve coefficient for G12 (degree 12 extension)
curve_order: int    # Order of the curve
field_modulus: int  # Prime modulus of the base field
twist           # Twist parameter for G2

def add(p1, p2):
    """
    Add two elliptic curve points.
    
    Args:
        p1: First point (G1 or G2)
        p2: Second point (G1 or G2, same type as p1)
        
    Returns:
        Point of same type as inputs representing p1 + p2
    """

def double(p):
    """
    Double an elliptic curve point.
    
    Args:
        p: Point to double (G1 or G2)
        
    Returns:
        Point of same type representing 2 * p
    """

def multiply(p, n):
    """
    Multiply a point by a scalar.
    
    Args:
        p: Point to multiply (G1 or G2)
        n (int): Scalar multiplier
        
    Returns:
        Point of same type representing n * p
    """

def neg(p):
    """
    Negate a point.
    
    Args:
        p: Point to negate (G1 or G2)
        
    Returns:
        Point of same type representing -p
    """

def eq(p1, p2) -> bool:
    """
    Test if two points are equal.
    
    Args:
        p1: First point
        p2: Second point
        
    Returns:
        bool: True if points are equal, False otherwise
    """

def is_inf(p) -> bool:
    """
    Check if a point is the point at infinity.
    
    Args:
        p: Point to check
        
    Returns:
        bool: True if point is at infinity, False otherwise
    """

def is_on_curve(p) -> bool:
    """
    Verify that a point lies on the curve.
    
    Args:
        p: Point to validate
        
    Returns:
        bool: True if point is on curve, False otherwise
    """

def pairing(p1, p2):
    """
    Compute the bilinear pairing e(p1, p2).
    
    Args:
        p1: Point in G1
        p2: Point in G2
        
    Returns:
        Element in GT (degree 12 extension field)
    """

def final_exponentiate(p):
    """
    Perform the final exponentiation step of pairing computation.
    
    Args:
        p: Element from the Miller loop computation
        
    Returns:
        Final pairing result in GT
    """

# Field element types (imported from py_ecc.fields)
FQ   # Base field element (Fp)
FQ2  # Quadratic extension field element (Fp2)
FQ12 # Degree 12 extension field element (Fp12, target group GT)
FQP  # General polynomial field element

from py_ecc.bls12_381 import G1, G2, multiply, add, is_on_curve

# Scalar multiplication
point1 = multiply(G1, 123)
point2 = multiply(G1, 456)

# Point addition
sum_point = add(point1, point2)

# Verify points are on curve
assert is_on_curve(point1)
assert is_on_curve(point2)
assert is_on_curve(sum_point)

# This should equal (123 + 456) * G1
expected = multiply(G1, 123 + 456)
assert eq(sum_point, expected)

from py_ecc.bls12_381 import G1, G2, multiply, pairing

# Create some points
p1 = multiply(G1, 123)
q1 = multiply(G2, 456)
p2 = multiply(G1, 789)
q2 = multiply(G2, 321)

# Compute pairings
pairing1 = pairing(p1, q1)
pairing2 = pairing(p2, q2)

# Pairing is bilinear: e(a*P, Q) = e(P, a*Q) = e(P, Q)^a
a = 42
pa_q = pairing(multiply(p1, a), q1)
p_aq = pairing(p1, multiply(q1, a))
p_q_a = pairing1 ** a

# All should be equal (up to field arithmetic precision)
print("Bilinearity verified")

from py_ecc.bls12_381 import G1, G2, multiply, pairing
from py_ecc.fields import bls12_381_FQ12 as FQ12

# Multiple pairing computation
points_g1 = [multiply(G1, i) for i in [1, 2, 3]]
points_g2 = [multiply(G2, i) for i in [4, 5, 6]]

# Compute individual pairings
pairings = [pairing(p1, p2) for p1, p2 in zip(points_g1, points_g2)]

# Product of pairings
result = FQ12.one()
for p in pairings:
    result = result * p

print(f"Multi-pairing result computed")

from py_ecc.bls12_381 import G1, G2, is_on_curve, is_inf, multiply
from py_ecc.fields import bls12_381_FQ as FQ

# Validate generator points
assert is_on_curve(G1)
assert is_on_curve(G2)
assert not is_inf(G1)
assert not is_inf(G2)

# Create and validate random points
random_point = multiply(G1, 12345)
assert is_on_curve(random_point)
assert not is_inf(random_point)

# Check point at infinity
zero_point = multiply(G1, 0)  # Should be point at infinity
assert is_inf(zero_point)

print("Point validation complete")

from py_ecc.fields import bls12_381_FQ as FQ, bls12_381_FQ2 as FQ2

# Work with field elements directly
a = FQ(123)
b = FQ(456)
c = a + b
d = a * b
e = a ** 3

# Quadratic extension field
f = FQ2([1, 2])  # 1 + 2*i where i^2 = -1
g = FQ2([3, 4])  # 3 + 4*i
h = f * g

print(f"Field arithmetic results: {c}, {d}, {e}")
print(f"Extension field result: {h}")

from typing import Optional, Tuple
from py_ecc.fields import bls12_381_FQ, bls12_381_FQ2

# Point types (None represents point at infinity)
Point2D = Optional[Tuple[bls12_381_FQ, bls12_381_FQ]]
Point2D_FQ2 = Optional[Tuple[bls12_381_FQ2, bls12_381_FQ2]]  # For G2 points

# Field element types
Field = bls12_381_FQ
ExtensionField2 = bls12_381_FQ2
ExtensionField12 = bls12_381_FQ12

from py_ecc.bls12_381 import is_on_curve, pairing, multiply, G1, G2

try:
    # Validate points before pairing
    p1 = multiply(G1, some_scalar)
    p2 = multiply(G2, some_other_scalar)
    
    if is_on_curve(p1) and is_on_curve(p2):
        result = pairing(p1, p2)
    else:
        raise ValueError("Invalid curve points")
        
except Exception as e:
    print(f"Curve operation error: {e}")