Executes the authentication process with the Telegram servers.

    :param sender: a connected `MTProtoPlainSender`.
    :return: returns a (authorization key, time offset) tuple.

async def do_authentication(sender):
    """
    Executes the authentication process with the Telegram servers.

    :param sender: a connected `MTProtoPlainSender`.
    :return: returns a (authorization key, time offset) tuple.
    """
    # Step 1 sending: PQ Request, endianness doesn't matter since it's random
    nonce = int.from_bytes(os.urandom(16), 'big', signed=True)
    res_pq = await sender.send(ReqPqMultiRequest(nonce))
    assert isinstance(res_pq, ResPQ), 'Step 1 answer was %s' % res_pq

    if res_pq.nonce != nonce:
        raise SecurityError('Step 1 invalid nonce from server')

    pq = get_int(res_pq.pq)

    # Step 2 sending: DH Exchange
    p, q = Factorization.factorize(pq)
    p, q = rsa.get_byte_array(p), rsa.get_byte_array(q)
    new_nonce = int.from_bytes(os.urandom(32), 'little', signed=True)

    pq_inner_data = bytes(PQInnerData(
        pq=rsa.get_byte_array(pq), p=p, q=q,
        nonce=res_pq.nonce,
        server_nonce=res_pq.server_nonce,
        new_nonce=new_nonce
    ))

    # sha_digest + data + random_bytes
    cipher_text, target_fingerprint = None, None
    for fingerprint in res_pq.server_public_key_fingerprints:
        cipher_text = rsa.encrypt(fingerprint, pq_inner_data)
        if cipher_text is not None:
            target_fingerprint = fingerprint
            break

    if cipher_text is None:
        raise SecurityError(
            'Step 2 could not find a valid key for fingerprints: {}'
            .format(', '.join(
                [str(f) for f in res_pq.server_public_key_fingerprints])
            )
        )

    server_dh_params = await sender.send(ReqDHParamsRequest(
        nonce=res_pq.nonce,
        server_nonce=res_pq.server_nonce,
        p=p, q=q,
        public_key_fingerprint=target_fingerprint,
        encrypted_data=cipher_text
    ))

    assert isinstance(
        server_dh_params, (ServerDHParamsOk, ServerDHParamsFail)),\
        'Step 2.1 answer was %s' % server_dh_params

    if server_dh_params.nonce != res_pq.nonce:
        raise SecurityError('Step 2 invalid nonce from server')

    if server_dh_params.server_nonce != res_pq.server_nonce:
        raise SecurityError('Step 2 invalid server nonce from server')

    if isinstance(server_dh_params, ServerDHParamsFail):
        nnh = int.from_bytes(
            sha1(new_nonce.to_bytes(32, 'little', signed=True)).digest()[4:20],
            'little', signed=True
        )
        if server_dh_params.new_nonce_hash != nnh:
            raise SecurityError('Step 2 invalid DH fail nonce from server')

    assert isinstance(server_dh_params, ServerDHParamsOk),\
        'Step 2.2 answer was %s' % server_dh_params

    # Step 3 sending: Complete DH Exchange
    key, iv = helpers.generate_key_data_from_nonce(
        res_pq.server_nonce, new_nonce
    )
    if len(server_dh_params.encrypted_answer) % 16 != 0:
        # See PR#453
        raise SecurityError('Step 3 AES block size mismatch')

    plain_text_answer = AES.decrypt_ige(
        server_dh_params.encrypted_answer, key, iv
    )

    with BinaryReader(plain_text_answer) as reader:
        reader.read(20)  # hash sum
        server_dh_inner = reader.tgread_object()
        assert isinstance(server_dh_inner, ServerDHInnerData),\
            'Step 3 answer was %s' % server_dh_inner

    if server_dh_inner.nonce != res_pq.nonce:
        raise SecurityError('Step 3 Invalid nonce in encrypted answer')

    if server_dh_inner.server_nonce != res_pq.server_nonce:
        raise SecurityError('Step 3 Invalid server nonce in encrypted answer')

    dh_prime = get_int(server_dh_inner.dh_prime, signed=False)
    g_a = get_int(server_dh_inner.g_a, signed=False)
    time_offset = server_dh_inner.server_time - int(time.time())

    b = get_int(os.urandom(256), signed=False)
    gb = pow(server_dh_inner.g, b, dh_prime)
    gab = pow(g_a, b, dh_prime)

    # Prepare client DH Inner Data
    client_dh_inner = bytes(ClientDHInnerData(
        nonce=res_pq.nonce,
        server_nonce=res_pq.server_nonce,
        retry_id=0,  # TODO Actual retry ID
        g_b=rsa.get_byte_array(gb)
    ))

    client_dh_inner_hashed = sha1(client_dh_inner).digest() + client_dh_inner

    # Encryption
    client_dh_encrypted = AES.encrypt_ige(client_dh_inner_hashed, key, iv)

    # Prepare Set client DH params
    dh_gen = await sender.send(SetClientDHParamsRequest(
        nonce=res_pq.nonce,
        server_nonce=res_pq.server_nonce,
        encrypted_data=client_dh_encrypted,
    ))

    nonce_types = (DhGenOk, DhGenRetry, DhGenFail)
    assert isinstance(dh_gen, nonce_types), 'Step 3.1 answer was %s' % dh_gen
    name = dh_gen.__class__.__name__
    if dh_gen.nonce != res_pq.nonce:
        raise SecurityError('Step 3 invalid {} nonce from server'.format(name))

    if dh_gen.server_nonce != res_pq.server_nonce:
        raise SecurityError(
            'Step 3 invalid {} server nonce from server'.format(name))

    auth_key = AuthKey(rsa.get_byte_array(gab))
    nonce_number = 1 + nonce_types.index(type(dh_gen))
    new_nonce_hash = auth_key.calc_new_nonce_hash(new_nonce, nonce_number)

    dh_hash = getattr(dh_gen, 'new_nonce_hash{}'.format(nonce_number))
    if dh_hash != new_nonce_hash:
        raise SecurityError('Step 3 invalid new nonce hash')

    if not isinstance(dh_gen, DhGenOk):
        raise AssertionError('Step 3.2 answer was %s' % dh_gen)

    return auth_key, time_offset