平成20年度電気関係学会四国支部連合大会
17-15384

A Chaotic Map with Variable Parameters to Design Cryptosystem
Shuichi Aono and Yoshifumi Nishio(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 iterating a chaotic map. These features look similar to the features of cryptology. For this reason, it is effective
to use chaotic maps for cryptosystems [1].
In this study, we propose a chaotic map that has a variable parameters. And we propose a cryptosystem using thisproposed chaotic map. A characteristic of the proposed cryptosystem is that different ciphertexts are generated from thesame plaintext.
2. Chaotic Map
In this study, we propose a modiﬁed chaotic map. The modiﬁed chaotic map is expressed as the following equation:
1X
(0 ≤ Xn ≤ α)
 Xn+1 = 1 − (1 − αn ) βFαβ :Xn −α β1
 X(α < Xn ≤ 1)
n+1 = 1 − ( 1−α )(1)where α and β are parameters changing the central coordinate and shape of the map.
The shape of the modiﬁed chaotic map is changed by combinations of α and β. Therefore, the feature of the generated
sequences is determined by these parameters. Figure 1 shows
the Lyapunov of the modiﬁed chaotic map that changes thevalue of β. We can see that the value of the Lyapunov exponent is a positive value in the range of β ∈ [0.2, 1.0].

The proposed cryptosystem uses three kind of keys, a public key, a private key and a shared secret key. The proposed
cryptosystem is composed of the following three parts.
3.1. Key generation
A decryptor sets an initial point for X0 and parameters αand β. Here, α and β are shared secret keys. In addition, set
the number of iterations A. This value A is a private key forAthe decryptor. XA = Fαβ(X0 ), namely, A-time iterations of
the modiﬁed chaotic map Fαβ are calculated. The decryptor
obtains X0 , XA as public keys, α, β as shared secret keys
and A as a private key.
3.2. Encryption process
A encryptor chooses the value B as a number of iterations,where B is an arbitrary value. The encryptor encrypts byusing a private key B. The encryption functions are describedas follows:B
C1 = M + Fαβ(XA ) = M + XA+BB
C2 = Fαβ (X0 ) = XB

(3)

where M is a plaintext.
(C1 , C2 ) are calculated. These values are sent to a receiveras ciphertexts.
3.3. Decryption process
In the decryption process, a decryptor calculates C1 −AFαβ(C2 ) by using the shared secret keys and the private key
A.
AA
C1 − Fαβ(C2 ) = M + XA+B − Fαβ(XB )= M + XA+B − XB+A=M(4)The plaintext M is decrypted. An important thing is that
there is no need to calculates the value of B.

Figure 1: Lyapunov exponent of modiﬁed chaotic map.
3. Proposed Cryptosystem
We propose a cryptosystem by using the following characteristics of a chaotic map.
BAA
BFαβ
◦ Fαβ(X0 ) = Fαβ◦ Fαβ(X0 )where A and B are the number of iterations.

4. Conclusions
In this study, we have proposed a modiﬁed chaotic map thathas a variable parameters. And we have proposed a cryptosystem using this chaotic map.
As the future subject, the security of the proposed cryptosystem will be investigated in more detail.

References[1] L. Kocarev, “Chaos-Based Cryptography : A Brief
Overview, ” IEEE Circuits and Systems Magazine, vol.
(2)
1, pp. 6-21, 2001.