"""Modular multiplicative inverse via extended Euclid.

modinv(a, m) returns x in [0, m) with a*x ≡ 1 (mod m), or None when
gcd(a, m) != 1 (no inverse exists). Handles negative a, m>1 only.
"""
def egcd(a, b):
    if b == 0:
        return (a, 1, 0)
    g, x1, y1 = egcd(b, a % b)
    return (g, y1, x1 - (a // b) * y1)

def modinv(a, m):
    if not isinstance(m, int) or m <= 1:
        raise ValueError("modulus must be int > 1")
    a = a % m
    g, x, _ = egcd(a, m)
    if g != 1 and g != -1:
        # normalize gcd sign
        if g < 0:
            g = -g
            x = -x
        if g != 1:
            return None
    x %= m
    # verify
    if (a * x) % m != 1:
        return None
    return x

if __name__ == "__main__":
    print(modinv(3, 11))