Nonlinear Dyn (2013) 74:419–427
DOI 10.1007/s11071-013-0979-4
ORIGINAL PAPER
Cryptanalysis and improvement of a three-party key
agreement protocol using enhanced Chebyshev polynomials
Fengjun Zhao ·Peng Gong ·Shuai Li ·
Mingguan Li ·Ping Li
Received: 19 April 2013 / Accepted: 10 June 2013 / Published online: 4 July 2013
© Springer Science+Business Media Dordrecht 2013
Abstract Three-party key agreement protocol is an
important cryptographic mechanism for secure com-
munication, which allows two parties authenticate
each other with the help of a trusted server. Very re-
cently, Lai et al.’s proposed a novel three-party key
agreement protocol using the enhanced Chebyshev
chaotic map and claimed their protocol could with-
stand various attacks. Unfortunately, in this paper, we
will show their protocol is vulnerable to the privi-
leged insider attack and the off-line password guess-
ing attack. To solve the problems, we propose an im-
proved three-party key agreement protocol using the
enhanced Chebyshev chaotic map. Security analysis
and performance analysis show our protocol not only
could withstand various attacks, but also has similar
performance. Therefore, it is very suitable for practi-
cal applications.
Keywords Authenticated key exchange ·
Three-party · Chebyshev polynomials · Security
F. Zhao · P. Gong (
) · S. Li · M. Li ·P. L i
National Key Laboratory of Mechatronic Engineering and
Control, School of Mechatronical Engineering, Beijing
Institute of Technology,
Beijing, China
e-mail: penggong@bit.edu.cn
1 Introduction
Characterized by sensitive dependence on initial con-
ditions, pseudo-randomness, and ergodicity, the Che-
byshev chaotic map is very suitable for the design
of cryptography, especially secret key cryptosystems.
Therefore, it has been used in the design of symmetric
encryption protocols [1–4], S-boxes [5, 6], and hash
functions [7].
Recently, the Chebyshev chaotic map is also used
to design the key agreement protocol. Key agreement
protocol plays an important role in modern secure
communication since it allows participants to authen-
ticate each other and generate a shared session key
future communication. Since the pioneering work of
Diffie and Hellman [8], many key agreement protocols
[9–15] have been proposed for different applications.
It is well known that the modular exponentiation op-
eration and scalar multiplication operation on elliptic
curves are complicated operation. Therefore, it is very
interesting to develop key agreement protocols using
the Chebyshev chaotic map since it has better perfor-
mance.
In 2007, Xiao et al. [16] proposed the first key
agreement protocol using the Chebyshev chaotic map.
However, Han [17] pointed out that Xiao et al.’s pro-
tocol is not secure at all. In 2009, Tseng et al. [18]
proposed an anonymous key agreement protocol using
the Chebyshev chaotic map. However, Niu and Wang
[19] pointed out Tseng et al.’s protocol suffers from
the insider attack. To improve security, Niu and Wang