2007 RISP International Workshop on Nonlinear Circuits and Signal Processing (NCSP'07)
Shanghai Jiao Tong University, Shanghai, China, Mar. 3-6, 2007

A User Authentication Protocol Using Chaotic Maps
Shuichi Aono† , Yoshifumi Nishio††

Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {aoichi,nishio}@ee.tokushima-u.ac.jp

Abstract
A chaotic map has sensitivity to a change in initial conditions and parameters, and a long-term forecast becomes
impossible by the iterations of a chaotic map. Thesefeatures look similar to the properties of the cryptology.
For that reason, it is effective to use chaotic maps forcryptosystems.
In this research, we propose an authentication protocol by three times of the authentication interaction.
This authentication protocol based on the iteration of
the coupled logistic maps in a public-key cryptography.
We investigate the vulnerability of this user authentication protocol.
1. Introduction
A chaotic map has various features. A chaotic map
has sensitivity to a change in initial conditions and parameters, and a long-term forecast becomes impossibleby the iterations of a chaotic map. These features look
similar to the properties of the cryptology. For that reason, it is effective to use chaotic maps for cryptosystems.
The chaotic cryptosystem is researched as an applicationof chaos in an engineering field [1-4].
A lot of chaotic cryptosystems that use the expansion
and the reduction of the chaotic map for the encryption
and decryption are reported [5][6]. On the other hand,
many researchers are reported about the disadvantagesof cryptosystem using the chaos map [7][8]. There isa possibility to be linear attacked when the piecewiselinear map is used in the cryptosystem. In addition, theadvantages of the chaotic map is demonstrated when thedirection of the expansion is used for the encryption. It
is undesirable to use reduced map for cryptosystem.
Moreover, a symmetric-key cryptography has theproblem of the delivery of the key distribution. This
problem can be solved by using a public-key cryptography. However, there are two problems in the applicationof the chaotic map to the public-key cryptography. One
is that the trap door is necessary to apply the publickey cryptosystem. It is difficult to develop trap door in
the chaotic maps. The other is that the equation form
is opened to the public. A chaotic map is shown by
the equation based on determinism. If the equation of
the chaotic map and their parameters are opened to thepublic, the behavior of the sequences is known.
A simplest but important thing is authentication technology in the cryptography. An authentication technology based on public-key cryptography has two patterns.
One is a digital signature. This authentication proves thecorrectness of the data. The other is a user authentication. This authentication proves the correctness of theuser. A zero-knowledge proof is an interactive method
for one party to prove to another that a statement istrue, without revealing anything other than the veracityof the statement. In a user authentication, the zeroknowledge proof is an ideal in safety, but the number of
the authentication interaction becomes large.
In this research, we use an authentication protocol by
three times of the authentication interaction. This authentication protocol based on iterations of the logistic
maps in a public-key cryptography. We investigate thevulnerability of this user authentication protocol.
2. Chaotic Map
A logistic map is one of the simplest chaotic maps.
The logistic map is expressed as the following equation:
Xn+1 = αXn (1 − Xn )(1)where α is a control parameter changing the behavior of
the generated sequence.
This map is shown in Fig. 1. Figure 2 shows an example of chaotic sequences generated by the logistic map.
The behavior like the uniform random number is seen in
α = 4.0.
In this research, we propose a user authentication protocol by using iterations of a chaotic map. We use coupled logistic maps as a chaotic map. The coupled logistic- 333 -Figure 1: Logistic map. (α = 4.0)
Figure 3: Coupled logistic maps. (α = 3.7, β = 0.06)

Figure 2: Chaotic sequence. (α = 4.0)
maps are expressed as follows; Xn+1 = αXn (1 − Xn ) + β(Yn − Xn )
Yn+1 = αYn (1 − Yn ) + β(Xn − Yn )Figure 4: Chaotic sequence of Xn . (α = 3.7, β = 0.006)(2)where α and β are parameters changing the behavior of
the sequences.
This map is shown in Fig. 3. Figures 4 and 5 show
the chaotic sequences of Xn and Yn . The generated sequences looks like the uniform random number, though
they have some bias. Figure 6 shows a relation between
the value of Xn and the value of Yn generated by thecoupled logistic maps.
In this research, we use parameters α = 3.7 and β =0.006. This map has sensitivity to a change in initialconditions X0 and Y0 . Therefore, a long-term forecastbecomes impossible by the iterations of the maps.
The coupled logistic maps are based on easy equations.

Figure 5: Chaotic sequence of Yn . (α = 3.7, β = 0.006)- 334 -

As for this protocol, safety has been proven though it
is not zero-knowledge proof. The safety of this protocol
is based on the difficulty of the factorization on prime
numbers problem and the discrete logarithm problem.
Therefore, this protocol is one of the ideal user authentication. Because this protocol is efficient and high safety.
Moreover, there is an advantage that there is a method of
conversion into a digital signature in the authentication
protocol by three times of the authentication interaction.

Figure 6: Relation between Xn and Yn
It is necessary to know the number of iterations of thecoupled logistic maps to obtain α and β. If the number of
iterations n is unknown, it is difficult to obtain correct
α and β from the value of X0 and Xn . In a similarway, it is very difficult to obtain correct X0 and Y0 fromthe value of Xn and Yn . Calculating the reverse-map
becomes solving the polynomial equation derived fromthe coupled logistic maps. The polynomial equation can
not be solved without knowing the correct values bothof Xn and Yn .

Figure 7: A user authentication protocol.

4. Proposed Authentication Protocol

3. A User Authentication Protocol
The encryption technology has the function of defending information, and becomes the element of other information security technologies like the digital signatureetc. A user authentication protocol is a simplest but
important technology in the cryptography. In a userauthentication, the zero-knowledge proof is an ideal insafety. This proof is an interactive method for one party
to prove to another that a statement is true, without
revealing anything other than the veracity of the statement.
However the zero-knowledge proof has a disadvantage
to implement. the number of the authentication interaction becomes large. The number of the interaction ofzero-knowledge proof must be four times or more. Then,the user authentication protocol of three times in thenumber of interactions has been proposed. This authentication protocol shown in Fig. 7.
Prover generated public key and private key, andopens the public key to the public. Prover and verifier interact three times between each other. Verifier
authenticates whether to accept or to reject.

In this research, we propose a user authentication protocol by using the coupled chaotic maps. This authentication protocol is composed of the following parts.
4.1. Key generation
A prover sets initial points X0 and Y0 of Eq. (1). Inaddition, set a number of iterations A1 , A2 . Here, Thevalue of A1 and A2 are k bits. Calculate (XA1 , YA1 ) =F A1 (X0 , Y0 ), namely A1 -time iterations of coupled logistic maps F . And calculate (XA2 , YA2 ) = F A2 (X0 , Y0 ),namely A2 -time iterations of coupled logistic maps F .
Also (XA , YA ) = (XA1 · XA2 , YA1 · YA2 )• private key : A1 , A2 , X0 , Y0 .
• public key : (XA , YA )4.2. Authentication protocol
A prover and a verifier interact three times in this user
authentication protocol.

- 335 -• step 1
A prover selects k bits numbers B1 and B2 .
A relationship between B1 and B2 as follows : A1 − B1 = A2 − B2or(3)

A1 − B2 = A2 − B1
where, (0 < B1 , B2 < A1 , A2 )Calculate (XB1 , YB1 ) and (XB2 , YB2 ). Send thesevalues to the verifier.
• step 2
A verifier chooses the k − 1 bit value C as an arbitrary value. Send this value C to the prover.
• step 3A prover calculates the value of D.
 D = A1 − B1 − Cor
D = A1 − B2 − C

References[1] L. Kocarev, “Chaos-Based Cryptography : A Brief
Overview, ” IEEE Circuits and Systems Magazine,
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(4)[5] T. Habutsu, Y. Nishio, I. Sasase and S. Mori, “ASecret Key Chaotic Map,” Trans. IEICE, vol. E73,no. 7, pp. 1041-1044, 1990.

This means that the prover uses the equation selected by step 1.

[6] M. Harada, Y. Nishio and A. Ushida, “A Cryptosystem Using Two Chaotic Maps,” Proceedings ofNOLTA’99, vol. 2, pp. 609-611, 1999.

• step 4[7] E. Biham, “Cryptanalysis of the chaotic-map crypA verifier confirms that the following equation istosystem suggested at EUROCRYPT’91,” Proc. Eusatisfied.
rocrypt ’91, pp. 532-534, 1991.
F D F C (XB1 , YB1 ) · F D F C (XB2 , YB2 ) = (XA , YA )(5)[8] N. Masuda, K. Aihara, “Cryptosystems with discretized chaotic maps,” IEEE Trans. Circuits andSystems I, vol. 49, pp. 28-40, 2002.

In this authentication protocol, the eavesdroppers can
not calculate correct keys from (XA , YA ). But, a prover
send the value of (XB1 , YB1 ) to the verifier. The verifier
can calculate the correct time series sequences after B1
iterations. There is a disadvantage that this authentication protocol is weak to the impersonation. Therefore, it
is a future subject to develop the authentication protocolthat does not send significant information.
5. Conclusions
In this research, we have proposed an authentication
protocol by three times of the authentication interaction.
This authentication protocol based on iterations of thecoupled logistic maps.
As the future subject, we investigate security of theproposed authentication protocol in more detail. And
we apply authentication protocols by using iterations ofa chaotic map to the digital signature.

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