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Uniformity of Ciphertext Generated by Two Chaotic Maps
Shuichi Aono† , Yoshifumi Nishio††Department of Electrical and Electronic Engineering
Tokushima University

1. Introduction
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. For that reason,
it is effective to use chaotic maps for cryptosystems. Thechaotic cryptosystem is researched as an application of chaosin an engineering field. On the other hand, the security of thechaotic cryptosystem is not investigated in detail. It is necessary to investigate the security and to clarify the vulnerabilityof the chaotic cryptosystem.
A cryptosystem using the tent map has been proposed [1].
A cryptosystem using two chaotic maps has been also proposed [2]. This system performed encryption and decryption
by using two chaotic maps, a skew tent map and a logisticmap, and their inverse maps. The method of repeating the encryption process twice has been proposed [3]. This method
used the same map as the first encryption and the second encryption.
In this research, we propose the method of repeating the encryption process twice by using different chaotic maps, andinvestigate the uniformity of the generated ciphertext.
2. Proposed Cryptosystem
The proposed method repeats the encryption process twice.
The encryption maps are shown in Figs. 1 and 2.

(a)

(b)

Figure 1: 1st encryption map F1 .

where F1 is the first encryption map and F2 is the secondencryption map. a and b are private keys in this cryptosystem.
In the first encryption process, set an initial point Y0 asa plaintext and X0 as a subtext, where, X0 is an arbitraryvalue. Calculate (Xn , Yn ) = F n (X0 , Y0 ), namely n-times
iterations of F1 .
In the second encryption process, set an initial point
X0! = Xn , where Xn is obtained by the first encryption process. Calculate n-times F (X0! , Y0! ), we can obtain Yn! fromF n (X0! , Y0! ) = (Xn! , Yn! ) as a ciphertext. And send this value
Yn! to the receiver.
The decryption function F −1 is a function that replaces awith b of the encryption function F .
3. Simulated Results
In this research, we investigate the correlation of the plaintext and the ciphertext. We use χ2 test for the uniformity.
This χ2 test examines the distribution of ciphertext encryptedwith different keys. The upper 5% of this χ2 is 16.9. If thevalue of χ2 test is 16.9 or less, the uniformity of the ciphertext is guaranteed.
The simulated results for 5 different plaintexts are shown in
Table 1.
Table 1: The results of the χ2 test.
Plaintext Conventional method
P1464.70
P2435.08P3489.06
P4451.06
P5544.06

Proposed method
7.90
7.28
11.20
13.82
10.50

4. Conclusions
In this research, we have investigated the uniformity of theciphertext. By simulated results, we confirmed that the distribution of the ciphertext is uniformity.
References[1] T. Habutsu, Y. Nishio, I. Sasase and S. Mori, “A Secret
Key Chaotic Map,” Trans. IEICE, vol. E73, no. 7, pp.
1041-1044, 1990.
[2] M. Harada, Y. Nishio and A. Ushida, “A Cryptosystem
Using Two Chaotic Maps,” Proceedings of NOLTA’99,
vol. 2, pp. 609-611, 1999.

(a)(b)

Figure 2: 2nd encryption map F2 .

[3] S. Aono, M. Wada and Y. Nishio, “Improvement of aCryptosystem Using Two Chaotic Maps,” Proceedings ofRISP International Workshop on Nonlinear Circuits andSignal Processing (NCSP’06), pp. 349-352, 2006.