A Cryptosystem Based on Iterations of Chaotic Mapwith Variable Parameter
Shuichi AONO and Yoshifumi NISHIO
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, 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 changes inthe initial conditions and parameters, and a long-term forecast becomes impossible by iterating a chaotic map. Thesefeatures look similar to the features of cryptology. For thisreason, it is eﬀective to use chaotic maps for cryptosystems.
In this research, we investigate features of the modiﬁed cut
map when the parameters αi are changed. We use the parameter αi as shared secret key in the cryptosystem. Hence,
it is important for the security of the proposed cryptosystem
to investigate the relation between the generated sequences
and the parameters. It inﬂuences the vulnerability of thecryptosystem.
1. Introduction
A chaotic map has sensitivity to a change in the initialconditions and parameters, and a long-term forecast becomes impossible by iterating a chaotic map. These features look similar to the features of cryptology. For thisreason, it is eﬀective to use chaotic maps for cryptosystems.
A chaotic cryptosystem is researched for the application ofchaos in engineering ﬁeld [1-4].
Many chaotic cryptosystems that use the expansion andthe reduction of chaotic maps for encryption and decryption are reported [5][6]. Figure 1 shows the basic chaoticcryptosystem. An encryptor encrypts plaintext by using theexpansion map, and a decryptor decrypts ciphertext by using the reduction map as a reverse-map. If we use one-to-nmap as the expansion map, the plaintext cannot be uniquelydecrypted, because calculating the reverse-map becomes nto-one map. Namely, it is necessary to propose the cryptosystem that one-to-one mapping is realized between theplaintext space and the ciphertext space in the decryption[7]. By using only an expansion map for encryption anddecryption, one-to-one mapping is realized simply. In addition, the advantages of the chaotic map is demonstrated
when the direction of the expansion is used for encryption.
It is undesirable to use reduced map for cryptosystems.
Recently, some researchers have proposed public-key
cryptography based on chaotic maps [8][9]. The delivery of the key distribution can be solved using public-key
cryptography. Though this is an important advantage of
public-key cryptography, public-key cryptography has an-other advantage. A diﬀerent ciphertext is generated fromthe same plaintext. An encryptor selects an arbitrary value
as a private key. The plaintext becomes the ciphertext that
depends on this private key. We consider that this featureis the interesting feature for the development of the chaotic
cryptosystem. Because, this feature means that the slightly
diﬀerent condition has to change in a randomly diﬀerent. Itmay be no exaggeration to say that this is a characteristicof the chaotic map.
In our previous research, we have proposed a modiﬁed
chaotic map that has a parameter changing the shape of themap and have proposed a cryptosystem using the modiﬁed
chaotic map [10]. A characteristic of this cryptosystem isthat diﬀerent ciphertexts are generated from the same plaintext.
In this research, we investigate features of the modiﬁed
cut map when the parameters αi are changed. We use theparameter αi as a shared secret key in the cryptosystem.
Hence, it is important for the security of the proposed cryptosystem to investigate the relation between the generated
sequences and the parameters. It inﬂuences the vulnerability of the cryptosystem.

Figure 1: Basic model of chaotic cryptosystem

2. Cryptosystem Based on Iterations of Map [10]We have proposed a cryptosystem by using iteration ofthe chaotic map. The simpliﬁed block diagram of the cryptosystem is shown in Fig. 2. The proposed cryptosystem iscomposed of the following three parts. The key generation,- 225 -the encryption process and the decryption process.

C1 = M + F B (XA ) = M + XA+B
C2 = F B (X0 ) = XB

(1)

where M is a plaintext.
(C1 , C2 ) are calculated. These values are sent to a receiver as ciphertexts.
• C1 , C2 : ciphertext.
• B : private key.
2.3. Decryption process
In the decryption process, a decryptor calculates C1 −F A (C2 ) by using a private key A.
C1 − F A (C2 ) = M + XA+B − F A (XB )= M + XA+B − XB+A=M(2)And decrypt the plaintext M. An important thing is that
there is no need to calculates the value of B. The decryptor uses only a shared secret key and a private key A fordecryption. By adopting the private, one-to-one mapping
between the encryption and the decryption is achieved.
Figure 2: Block diagram of cryptosystem
3. Chaotic Map
2.1. Key generation
This cryptosystem uses three kind of keys, a public key,a private key and a shared secret key. A shared secret key
means private key between the decryptor and the encryptor.
And, the keys are decided by the decryptor. The process ofkey generation is as follows.
A decryptor sets an initial point for X0 and a parameter
α. Here, α is a shared secret key. In addition, set the number of iterations A. This value A is a private key for thedecryptor. The decryptor keep a private key A to himself.
XA = F A (X0 ), namely, A-time iterations of the modiﬁed
chaotic map F are calculated.
The decryptor can obtain X0 , XA as public keys, α as ashared secret key and A as a private key.
• X0 , XA : public key.
• α : shared secret key.
• A : private key.
2.2. Encryption process
We explain an encryption process of this cryptosystem.
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 described as follows :In this research, we use a modiﬁed cut map. The modiﬁed cut map is expressed as the following equation:1Xn+1 =Xnα1

1α1

Xn+1 =Xn −


α−αα
212 − α1
.
.
F:
.



1αk−1

Xn+1 =Xn −


α−αα
kk−1k − αk−1



1αk

 Xn+1 =Xn −1 − αk1 − αk(0 ≤ Xn ≤ α1 )(α1 < Xn ≤ α2 )..
.
(αk−1 < Xn ≤ αk )(αk < Xn ≤ 1)

(3)
where αi(i=1,...,k) is a parameter changing the shape of themap.
The shape of the modiﬁed chaotic map is changed by αi .
Therefore, the feature of the generated sequences is determined by this parameter. The modiﬁed cut map becomes a
original cut map at the case of k = 1 and α1 = 0.5. Thismap is shown in Fig. 3.
Next, we investigate the Lyapunov exponent of the modiﬁed cut map. The Lyapunov exponent is expressed as thefollowing equation.

- 226 -1 ∑log |F ′ (Xk )|N→∞ N
k=0N−1

λ = lim(4)

2. Calculate Xn with diﬀerent initial values. Make an l×l
contingency table fi .
3. Calculateχ =2()2l∑
fi − fi′
i=1fi′(5)where fi′ are ideal.
The upper 5% of this χ2 is 16.9. If χ2 is 16.9 or less, theuniformity of the generated sequence is guaranteed. The
simulated result with changing the parameter k is shownin Fig. 5. From this ﬁgure, we conﬁrm that almost thedistributions of Xn are uniform. In other words, we can use
any combination numbers k for the proposed cryptosystembecause the distribution of Xn is uniform and independent
on the initial values of the modiﬁed cut map.

Figure 3: Modiﬁed cut map (k = 1, α1 = 0.5)
Figure 4 shows the Lyapunov of the modiﬁed cut map
that changes the value of k. The horizontal axis shows thevalue of k, and the vertical axis shows the value of the Lyapunov exponent.

Figure 5: χ2 test for uniformity

4.2. Number of iterations
Figure 4: Lyapunov exponent of modiﬁed cut map

4. Simulated Results
In this research, we investigate the characteristic of themodiﬁed cut map. It is important for the security of thechaotic cryptosystem to investigate the relation between thegenerated sequences and the parameter.
4.1. Uniformity of generated sequences
We investigate the correlation of X0 and Xn . We use theχ2 test for the uniformity. The χ2 test for uniformity isperformed as follows.
1. Divide the interval [0, 1] into l class intervals.

Next, we investigate numbers of iterations A and B. The
numbers of iterations correspond to the key sensitivity. The
modiﬁed cut map has sensitivity to changes in parametersαi . Small variations of the keys produce large variations byiterating the map. Therefore, cipher-breaking becomes difﬁcult by increasing numbers of iterations. However, if thenumber of iteration is small, there are no large diﬀerences
between the variations of the keys. It is undesirable to use
as keys.
In order to determine the number of iterations, we simulated the sequences with slightly diﬀerent keys. Figure
6 (a) shows the diﬀerence of the generated value Xn withk = 1 between the case of X0 and the case of X0 + 10−40 .
The horizontal axis shows the number of iterations, and thevertical axis shows the vertical axis shows the diﬀerence
between the generated sequences. And, the diﬀerence ofthe two values cases of k = 2 and k = 3 are shown in Fig.
6 (b)(c).
From these ﬁgures, there are no large diﬀerences unlessthe number of iteration is small. It is undesirable to use this- 227 -range as keys. We can see the diﬀerence between the twovalues that have changed nonuniformity after several iterations. In addition, we can see that there is inverse relationship between the number of k and the number of iterations.
Namely, we can easily obtain diﬀused sequences by iterating a simple chaotic map and the fast diﬀusion is realized
by using the combination map.
5. Conclusions
In this research, we have investigated features of themodiﬁed cut map when the parameters αi are changed. We
have conﬁrmed that there is inverse relationship for diﬀusing the plaintext between the number of k and the numberof iterations.
As the future subject, the security of the proposed cryptosystem will be investigated in more detail. We will develop the secret sharing scheme by using iterations of achaotic map.

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(b)

(c)
Figure 6: Diﬀerence between two values for (a) k = 1, (b)k = 2, (c) k = 3- 228 -