How to use the ecdsa.numbertheory.inverse_mod function in ecdsa

http://www.secg.org/sec1-v2.pdf
        '''
        curve = ecdsa.SECP256k1.curve
        G = ecdsa.SECP256k1.generator
        order = ecdsa.SECP256k1.order
        yp = (i %2)
        r, s = ecdsa.util.sigdecode_string(signature, order)
        x = r + (i // 2 ) * order
        alpha = ((x * x * x) + (curve.a() * x) + curve.b()) % curve.p()
        beta = ecdsa.numbertheory.square_root_mod_prime(alpha, curve.p())
        y = beta if (beta - yp) % 2 == 0 else curve.p() - beta
        # generate R
        R = ecdsa.ellipticcurve.Point(curve, x, y, order)
        e = ecdsa.util.string_to_number(digest)
        # compute Q
        Q = ecdsa.numbertheory.inverse_mod(r, order) * (s * R + (-e % order) * G)
        # verify message
        if not ecdsa.VerifyingKey.from_public_point(Q, curve=ecdsa.SECP256k1).verify_digest(signature, digest,
                                                                                            sigdecode=ecdsa.util.sigdecode_string) :
            return None
        return ecdsa.VerifyingKey.from_public_point(Q, curve=ecdsa.SECP256k1)

http://www.secg.org/sec1-v2.pdf
        '''
        curve = ecdsa.SECP256k1.curve
        G = ecdsa.SECP256k1.generator
        order = ecdsa.SECP256k1.order
        yp = (i %2)
        r, s = ecdsa.util.sigdecode_string(signature, order)
        x = r + (i // 2 ) * order
        alpha = ((x * x * x) + (curve.a() * x) + curve.b()) % curve.p()
        beta = ecdsa.numbertheory.square_root_mod_prime(alpha, curve.p())
        y = beta if (beta - yp) % 2 == 0 else curve.p() - beta
        # generate R
        R = ecdsa.ellipticcurve.Point(curve, x, y, order)
        e = ecdsa.util.string_to_number(digest)
        # compute Q
        Q = ecdsa.numbertheory.inverse_mod(r, order) * (s * R + (-e % order) * G)
        # verify message
        if not ecdsa.VerifyingKey.from_public_point(Q, curve=ecdsa.SECP256k1).verify_digest(signature, digest,
                                                                                            sigdecode=ecdsa.util.sigdecode_string) :
            return None
        return ecdsa.VerifyingKey.from_public_point(Q, curve=ecdsa.SECP256k1)

def recover_public_key(self, digest, signature, i):
        curve = ecdsa.SECP256k1.curve
        G = ecdsa.SECP256k1.generator
        order = ecdsa.SECP256k1.order
        yp = (i % 2)
        r, s = ecdsa.util.sigdecode_string(signature, order)
        x = r + (i // 2) * order
        alpha = ((x * x * x) + (curve.a() * x) + curve.b()) % curve.p()
        beta = ecdsa.numbertheory.square_root_mod_prime(alpha, curve.p())
        y = beta if (beta - yp) % 2 == 0 else curve.p() - beta
        R = ecdsa.ellipticcurve.Point(curve, x, y, order)
        e = ecdsa.util.string_to_number(digest)
        Q = ecdsa.numbertheory.inverse_mod(r, order) * (s * R +
                                                        (-e % order) * G)
        if not ecdsa.VerifyingKey.from_public_point(
                Q, curve=ecdsa.SECP256k1).verify_digest(
                    signature, digest, sigdecode=ecdsa.util.sigdecode_string):
            return None
        return ecdsa.VerifyingKey.from_public_point(Q, curve=ecdsa.SECP256k1)

def test_inverse(curve, r):
    x = r.randrange(1, curve.p)
    y = int2bn(x)
    lib.bn_inverse(y, int2bn(curve.p))
    y = bn2int(y)
    y_ = ecdsa.numbertheory.inverse_mod(x, curve.p)
    assert y == y_

# 1.1
        x = r + (recid//2) * order
        # 1.3
        alpha = ( x * x * x  + curveFp.a() * x + curveFp.b() ) % curveFp.p()
        beta = msqr.modular_sqrt(alpha, curveFp.p())
        y = beta if (beta - recid) % 2 == 0 else curveFp.p() - beta
        # 1.4 the constructor checks that nR is at infinity
        try:
            R = Point(curveFp, x, y, order)
        except:
            raise InvalidECPointException()
        # 1.5 compute e from message:
        e = string_to_number(h)
        minus_e = -e % order
        # 1.6 compute Q = r^-1 (sR - eG)
        inv_r = numbertheory.inverse_mod(r,order)
        try:
            Q = inv_r * ( s * R + minus_e * G )
        except:
            raise InvalidECPointException()
        return klass.from_public_point( Q, curve )

order = G.order()
        # extract r,s from signature
        r, s = util.sigdecode_string(sig, order)
        # 1.1
        x = r + (recid//2) * order
        # 1.3
        alpha = ( x * x * x  + curveFp.a() * x + curveFp.b() ) % curveFp.p()
        beta = msqr.modular_sqrt(alpha, curveFp.p())
        y = beta if (beta - recid) % 2 == 0 else curveFp.p() - beta
        # 1.4 the constructor checks that nR is at infinity
        R = Point(curveFp, x, y, order)
        # 1.5 compute e from message:
        e = string_to_number(h)
        minus_e = -e % order
        # 1.6 compute Q = r^-1 (sR - eG)
        inv_r = numbertheory.inverse_mod(r,order)
        Q = inv_r * ( s * R + minus_e * G )
        return klass.from_public_point( Q, curve )

def curve25519_affine_double(x1, y1):
    if (x1, y1) == (1, 0):
        return (1, 0)
    
    x2 = (3 * x1 ** 2 + 2 * CURVE_A * x1 + 1) ** 2 * inverse_mod((2 * y1) ** 2, CURVE_P) - CURVE_A - x1 - x1
    y2 = (2 * x1 + x1 + CURVE_A) * (3 * x1 ** 2 + 2 * CURVE_A * x1 + 1) * inverse_mod(2 * y1, CURVE_P) - \
        (3 * x1 ** 2 + 2 * CURVE_A * x1 + 1) ** 3 * inverse_mod((2 * y1) ** 3, CURVE_P) - y1
    return x2 % CURVE_P, y2 % CURVE_P

order = G.order()
        # extract r,s from signature
        r, s = util.sigdecode_string(sig, order)
        # 1.1
        x = r + (recid / 2) * order
        # 1.3
        alpha = (x * x * x + curveFp.a() * x + curveFp.b()) % curveFp.p()
        beta = msqr.modular_sqrt(alpha, curveFp.p())
        y = beta if (beta - recid) % 2 == 0 else curveFp.p() - beta
        # 1.4 the constructor checks that nR is at infinity
        R = Point(curveFp, x, y, order)
        # 1.5 compute e from message:
        e = string_to_number(h)
        minus_e = -e % order
        # 1.6 compute Q = r^-1 (sR - eG)
        inv_r = numbertheory.inverse_mod(r, order)
        Q = inv_r * (s * R + minus_e * G)
        return cls.from_public_point(Q, curve)

compressed = False
    recid = nV - 27
    # 1.1
    x = r + (recid/2) * order
    # 1.3
    alpha = ( x * x * x  + curve.a() * x + curve.b() ) % curve.p()
    beta = modular_sqrt(alpha, curve.p())
    y = beta if (beta - recid) % 2 == 0 else curve.p() - beta
    # 1.4 the constructor checks that nR is at infinity
    R = ellipticcurve.Point(curve, x, y, order)
    # 1.5 compute e from message:
    h = Hash( msg_magic( message ) )
    e = string_to_number(h)
    minus_e = -e % order
    # 1.6 compute Q = r^-1 (sR - eG)
    inv_r = numbertheory.inverse_mod(r,order)
    Q = inv_r * ( s * R + minus_e * G )
    public_key = ecdsa.VerifyingKey.from_public_point( Q, curve = SECP256k1 )
    # check that Q is the public key
    public_key.verify_digest( sig[1:], h, sigdecode = ecdsa.util.sigdecode_string)
    # check that we get the original signing address
    addr = public_key_to_bc_address(encode_point(public_key, compressed))
    if address == addr:
        return True
    else:
        #print addr
        return False