Pell Curve Finite Field Group Structure

Pell.py

# Self-contained Python script for Pell conic group law check

def pell_add(a, b, D, p):
    """Group law on x^2 - D*y^2 = 1 mod p."""
    x1, y1 = a
    x2, y2 = b
    x3 = (x1 * x2 + D * y1 * y2) % p
    y3 = (x1 * y2 + x2 * y1) % p
    return x3, y3

def pell_check(point, D, p):
    """Verify that point satisfies x^2 - D*y^2 ≡ 1 mod p."""
    x, y = point
    lhs = (x*x - D*y*y) % p
    return lhs == 1 % p

if __name__ == "__main__":
    # Parameters
    D = 16991
    p = 20959
    P = (2611, 15590)
    invP = (P[0], (-P[1]) % p)
    Q = invP
    print("Inverse of P:", invP)

    # Check the point is on the curve
    print("Checking that P is on the Pell conic:")
    print("Valid point:", pell_check(P, D, p))

    # Double the point
    P2 = pell_add(P, Q, D, p)
    print("\nP + Q =", P2)

    # Check if result is identity (1, 0)
    print("\nIs P+P the identity (1,0)?", P2 == (1, 0))

    # Verify it also lies on the conic
    print("P+P on the conic:", pell_check(P2, D, p))