How to use the ecdsa.util.sigdecode_string function in ecdsa

def recover_public_key(digest, signature, i, message=None):
    """ Recover the public key from the the signature
    """

    # See http: //www.secg.org/download/aid-780/sec1-v2.pdf section 4.1.6 primarily
    curve = ecdsa.SECP256k1.curve
    G = ecdsa.SECP256k1.generator
    order = ecdsa.SECP256k1.order
    yp = i % 2
    r, s = ecdsa.util.sigdecode_string(signature, order)
    # 1.1
    x = r + (i // 2) * order
    # 1.3. This actually calculates for either effectively 02||X or 03||X depending on 'k' instead of always for 02||X as specified.
    # This substitutes for the lack of reversing R later on. -R actually is defined to be just flipping the y-coordinate in the elliptic curve.
    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
    # 1.4 Constructor of Point is supposed to check if nR is at infinity.
    R = ecdsa.ellipticcurve.Point(curve, x, y, order)
    # 1.5 Compute e
    e = ecdsa.util.string_to_number(digest)
    # 1.6 Compute Q = r^-1(sR - eG)
    Q = ecdsa.numbertheory.inverse_mod(r, order) * (s * R + (-e % order) * G)

    if SECP256K1_MODULE == "cryptography" and message is not None:
        if not isinstance(message, bytes):

def _recover_key(self, digest, signature, i) :
        ''' Recover the public key from the sig
            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 test_truncated_string_signatures(verifying_key, hash_alg, signature, trunc):
    # check if a malformed string encoded signature causes the same exception
    # to be raised irrespective of the type of error
    sig = bytearray(signature)
    sig = sig[:trunc]
    sig = binary_type(sig)

    try:
        verifying_key.verify(sig, example_data, getattr(hashlib, hash_alg),
                             sigdecode_string)
        assert False
    except BadSignatureError:
        assert True

def verify_message(address, signature, message):
    """ See http://www.secg.org/download/aid-780/sec1-v2.pdf for the math """
    curve = ecdsa.curves.SECP256k1.curve  # curve_secp256k1
    G = ecdsa.curves.SECP256k1.generator
    order = G.order()
    # extract r,s from signature
    if len(signature) != 65: raise BaseException("Wrong signature")
    r, s = util.sigdecode_string(signature[1:], order)
    nV = ord(signature[0])
    if nV < 27 or nV >= 35:
        raise BaseException("Bad encoding")
    if nV >= 31:
        compressed = True
        nV -= 4
    else:
        compressed = False

    recid = nV - 27
    # 1.1
    x = r + (recid / 2) * order
    # 1.3
    y = tools.point_y_from_x(x, recid % 2 > 0)
    # 1.4 the constructor checks that nR is at infinity
    R = ellipticcurve.Point(curve, x, y, order)

curve = SECP256k1.curve
    G = SECP256k1.G
    order = SECP256k1.order

    secret, compressed = private_key_to_secret_check_compressed(private_key)
    if secret == None:
        return None

    h = double_sha256(format_message_for_signing(message))
    signing_key = ecdsa.SigningKey.from_secret_exponent(secret, curve=SECP256k1.ecdsa_curve)
    ecdsa_signature = signing_key.sign_digest(h, sigencode=ecdsa.util.sigencode_string)
    
    public_point = signing_key.get_verifying_key().pubkey.point

    e = ecdsa.util.string_to_number(h)
    r, s = ecdsa.util.sigdecode_string(ecdsa_signature, order)

    # Okay, now we have to guess and check parameters for j and yp
    found = False
    for j in range(SECP256k1.h + 1):
        x = r + j * order
        alpha = ((x * x * x) + (curve.a() * x) + curve.b()) % curve.p()
        beta = ecdsa.numbertheory.square_root_mod_prime(alpha, curve.p())
        for yp in range(2):
            if (beta - yp) % 2 == 0:
                y = beta
            else:
                y = curve.p() - beta
            R = ecdsa.ellipticcurve.Point(curve, x, y, order)
            Q = ecdsa.numbertheory.inverse_mod(r, order) * (s * R + (-e % order) * G)
            if Q == public_point:
                found = True

def from_signature(klass, sig, recid, h, curve):
        """ See http://www.secg.org/download/aid-780/sec1-v2.pdf, chapter 4.1.6 """
        from ecdsa import util, numbertheory
        import msqr
        curveFp = curve.curve
        G = curve.generator
        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 )

# in the elliptic curve.
        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
        # 1.4 Constructor of Point is supposed to check if nR is at infinity.
        R = ecdsa.ellipticcurve.Point(curve, x, y, order)
        # 1.5 Compute e
        e = ecdsa.util.string_to_number(digest)
        # 1.6 Compute Q = r^-1(sR - eG)
        Q = ecdsa.numbertheory.inverse_mod(r, order) * (s * R +
                                                        (-e % order) * G)
        # Not strictly necessary, but let's verify the message for
        # paranoia's sake.
        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 from_signature(klass, sig, recid, h, curve):
        """ See http://www.secg.org/download/aid-780/sec1-v2.pdf, chapter 4.1.6 """
        from ecdsa import util, numbertheory
        from . import msqr
        curveFp = curve.curve
        G = curve.generator
        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 sign(self, msg_hash):
        private_key = MySigningKey.from_secret_exponent(self.secret, curve=SECP256k1)
        public_key = private_key.get_verifying_key()
        signature = private_key.sign_digest_deterministic(msg_hash, hashfunc=hashlib.sha256,
                                                          sigencode=ecdsa.util.sigencode_string)
        assert public_key.verify_digest(signature, msg_hash, sigdecode=ecdsa.util.sigdecode_string)
        return signature

def from_signature(klass, sig, recid, h, curve):
        """ See http://www.secg.org/download/aid-780/sec1-v2.pdf, chapter 4.1.6 """
        from ecdsa import util, numbertheory
        from . import msqr
        curveFp = curve.curve
        G = curve.generator
        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 )