2008 International Symposium on Nonlinear Theory and its Applications
NOLTA'08, Budapest, Hungary, September 7-10, 2008

Chaotic Map with Parameter Changing Shape of the Map for a Cryptosystem
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 in
the 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 propose a chaotic map withparameter changing the shape of the map. And we propose a cryptosystem using a chaotic map. A characteristicof the proposed cryptosystem is that diﬀerent ciphertexts
are generated from the same plaintext. The ciphertext depends on the private key of the encryptor. This cryptosystem is a symmetric-key cryptography that has an advantageof public-key cryptography. We investigate the vulnerability of this cryptosystem.
1. Introduction
A chaotic map has sensitivity to changes 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[1][2]. A chaotic cryptosystem is researched for theapplication of chaos in engineering ﬁeld [3]-[9].
Some chaotic cryptosystems that use the expansion andthe reduction of chaotic maps for encryption and decryption are reported [5][6]. Figure 1 shows the basic chaotic
cryptosystem. An encryptor encrypts plaintext by using the
expansion map, and a decryptor decrypts ciphertext by using its reverse-map. If we use one-to-n map as the expansion map, the plaintext cannot be uniquely decrypted, because calculating the reverse-map becomes n-to-one map.
Namely, it is necessary to propose the cryptosystem that
one-to-one mapping is realized between the plaintext space
and the ciphertext space in the decryption [7]. By using
only an expansion map for encryption and decryption, oneto-one mapping is realized simply. In addition, the advantages of the chaotic map is demonstrated when the directionof the expansion is used for encryption. It is undesirable touse 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. It
may be no exaggeration to say that this is a characteristicof the chaotic map.
In this research, we propose a chaotic map that has a parameter changing the shape of the map. And we proposea cryptosystem using this proposed chaotic map. A characteristic of the proposed cryptosystem is that a diﬀerent
ciphertexts are generated from the same plaintext. The ciphertext depends on the public key of the encryptor. This
characteristic is one of the important features in public-key
cryptography. This cryptosystem is symmetric-key cryptography that has a characteristic of public-key cryptography. We investigate the vulnerability of this cryptosystem.

Figure 1: Basic chaotic cryptosystem

2. Chaotic Map
In this research, we propose a modiﬁed chaotic map. The
modiﬁed chaotic map is expressed as the following equation:

Xn 1



Xn+1 = 1 − (1 − ) β



α


F:



Xn − α β1) Xn+1 = 1 − (1−α- 384 -

(0 ≤ Xn ≤ α)(1)(α < Xn ≤ 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. For
example, the modiﬁed chaotic map becomes a tent map at
the case of α = 0.5 and β = 1.0. In the case of α = 0.5 andβ = 0.5, the modiﬁed chaotic map becomes a logistic map.
These maps are shown in Fig. 2.

the Lyapunov exponent is a positive value in the range ofβ ∈ [0.2, 1.0].

Figure 3: Lyapunov exponent of modiﬁed chaotic map.
The modiﬁed chaotic map has sensitivity to a change inparameters α, β and the initial state X0 . Therefore, a longterm forecast becomes impossible by iterating the map.
However, the modiﬁed chaotic map is expressed by simple equations. If the number of iterations n and parameters
α and β are known, it is easy to calculate Xn from initialvalue X0 . In other words, it is necessary to know both thenumber of iterations and parameters to calculate the correct
Xn . Similarly, it is diﬃcult to calculate parameters α, β andn from X0 and Xn . In this research, we use parameters α
and β as a shared secret keys.

(a)

3. Proposed Cryptosystem
We propose a cryptosystem by using iteration of themodiﬁed chaotic map. In order to adopt the private key forsymmetric-key cryptography, the chosen private key must
be negated by the decryptor in the decryption process. We
use the following characteristics of a chaotic map to solve
this problem.
(b)

F B ◦ F A (X0 ) = F A ◦ F B (X0 )Figure 2: Modiﬁed chaotic map (α = 0.5, (a) β = 1.0, (b)β = 0.5)
Next, we investigate the Lyapunov exponent of the modiﬁed chaotic map. The Lyapunov exponent is expressed as
the following equation.
1 ∑log |F ′ (Xk )|λ = lim
N→∞ N
k=0
N−1

(2)

Figure 3 shows the Lyapunov of the modiﬁed chaotic
map that changes the value of β. The horizontal axis shows
the value of β, and the vertical axis shows the value ofthe Lyapunov exponent. We can see that the value of(3)

where A and B are the number of iterations.
The proposed cryptosystem is composed of the following three parts. The key generation, the encryption process
and the decryption process.
3.1. 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 a secret key between the decryptor and the encryptor. And, the keys are decided by the decryptor. The process of key generation is as follows.
A decryptor sets an initial point for X0 , parameters α andβ. Here, α and β are shared secret keys. In addition, set the
number of iterations A. This value A is a private key for the- 385 -

decryptor. The decryptor keeps a private key A secretly.
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, β asshared secret keys and A as a private key.
• X0 , XA : public key.
• α, β : shared secret key.
• A : private key.
3.2. 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 :C1 = M + F B (XA ) = M + XA+B
C2 = F B (X0 ) = XB

(4)

where M is a plaintext.
(C1 , C2 ) are calculated. These values are sent to a receiver as ciphertexts.
• C1 , C2 : ciphertext.
• B : private key.
3.3. 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

(5)

The plaintext M is decrypted. In this cryptosystem, we
use only the expansion map for encryption and decryption
by adopting the private keys. An important thing is that
there is no need to calculates the value of B. The decryptor
uses only the shared secret keys α, β and the private key A
for decryption.
4. Security Analysis
4.1. Shared secret key space size
The shared secret key space size is required over 128
bits (40 digits) for the defense against brute force attack. A
brute force attack is a method of defeating a cryptographic
scheme by trying all possible keys. We determined the key
space sizes as follows :• α ∈ [0, 1] : 40 digits, β ∈ [0.2, 1.0] : 10 digits.

In the proposed cryptosystem, the modiﬁed chaotic map
is changed by combinations of α and β. Namely, the key
space size of this cryptosystem equals the key space of α
times the key space of β. If there is no eﬀective shortcut
attack, the key space size is computationally-secure key
space. Next, we consider the rounding error by the discrete chaotic map. We truncate Xn to 40 decimal placesby iterating of the map. Therefore, the chaos space is certainly reduced. However, the reduction of the chaos spacedoes not aﬀect the key space. We secure enough key spacein consideration of rounding error. We use a GNU Multiple Precision Arithmetic Library (GMP) [10] to implement
the proposed cryptosystem. GMP is a library for arbitrary
precision arithmetic, operating on signed integers, rational
numbers, and ﬂoating point numbers.
4.2. Private key space size
We investigate the numbers of iterations A and B as theprivate keys. The security of the proposed cryptosystem depends on the parameters α and β. Hence, it is not necessary
to ensure the large key space for A and B against the brute
force attack. However, the number of iterations should belarge enough to ensure α and β sensitivity. The modiﬁed
chaotic map has a sensitivity to a change in the parameter.
Small variations of the parameter produce large variations
by iterating the map. Therefore, cipher-breaking becomesdiﬃcult by increasing the numbers of iterations. However,if the number of iteration is small, there are no large diﬀerences between the variations of the parameters. It is undesirable to use as small number of iterations as keys.
In order to determine the number of iterations, we simulated the sequences generated with slightly diﬀerent keys.
Figure 4 (a) shows the diﬀerence of the generated values
Xn between the case of α and the case of α + 10−40 . The
horizontal axis shows the value of α, and the vertical axis
shows the ﬁrst number of iterations when the diﬀerence between two values becomes larger than 0.5. And, the diﬀerence of the two values between the case of β and the caseof β + 10−10 is shown in Fig. 4 (b). From this ﬁgure, we
determine that the number of iterations is larger than 500.
4.3. Public key consideration
The proposed cryptosystem uses the public-key cryptography for a part of the cryptosystem, and the number of thedelivery of keys are decreased. There is a possibility that allof the pair of parameters α and β generates the value of XA
after N-times iterations. Because the generated sequencesby the modiﬁed chaotic map are a long-term sequences.
Surely, we secure enough key space size in considerationof brute force attack. The eavesdropper cannot decide thecorrect key and does not know whether the obtained key iscorrect key or not, even if the eavesdroppers search all of
the keys by brute force attack.
For that reason, It is diﬃcult for eavesdroppers to obtain correct α, β and A from the value of X0 and XA . We

- 386 -

think that this feature is an advantage compared with theconventional chaotic symmetric-key cryptosystem. The security of the cryptosystem becomes safer by setting the different private key at the encryption.
5. Conclusions
In this research, we have proposed a modiﬁed chaotic
map with parameter changing the shape of the map. And
we have proposed a cryptosystem using this map. A characteristic of the proposed cryptosystem is that a diﬀerent
ciphertext is generated from the same plaintext. We haveinvestigated vulnerability of this cryptosystem.
As the future subject, the security of the proposed cryptosystem will be investigated in more detail. In particular,
independent and identically distributed of the generated sequences. The sequences generated by the modiﬁed chaotic
map is not uniform for some parameter β. Therefore, weneed to adopt the protocol such as using the lower bit ofgenerated sequences. We will develop the secret sharing
scheme by using iterations of a chaotic map.

(a)References(b)
Figure 4: Diﬀerence between two values for (a) α and α +10−40 , (b) β and β + 10−10thought that the values of α, β and the value of A did not becalculated even if the value of X0 and the value of XA wereopened to the public.
4.4. Chosen plaintext attack
Next, we consider a chosen plaintext attack (CPA) for
this cryptosystem. A CPA is an attack model for cryptanalysis which presumes that the attacker has the capability tochoose arbitrary plaintexts to be encrypted and obtain thecorresponding ciphertexts.
The proposed cryptosystem is a simple but eﬀective system against the CPA. Though an eavesdropper can easily
obtain two values XA+B and XB from a pair of plaintext andits ciphertexts by using the diﬀerence between the two ciphertexts, it is diﬃcult for the eavesdropper to obtain correct A, B, α, β from XA , XB and XA+B . For this reason, we
consider that the CPA is not eﬀective compared with thebrute force attack.
In addition, the encryptor can easily change the numberof iterations B at the every encryption. The encryptor can
generate a diﬀerent ciphertext from the same plaintext. We[1] L. Kocarev, “Chaos-based cryptography : a brief
overview, ” IEEE Circuits and Systems Magazine, vol.
1, pp. 6-21, 2001.
[2] N. Masuda, G. Jakimoski, K. Aihara and L. Kocarev,“Chaotic block ciphers : from theory to practical algorithms,” IEEE Trans. Circuits and Systems I, vol. 53,
no. 6, pp. 1341-1352, 2006.
[3] G. Jakimoski and L. Kocarev, “Chaos and cryptography : block encryption ciphers based on chaotic maps,”
IEEE Trans. Circuits and Systems I, vol. 48, no. 2, pp.
163-169, 2001.
[4] X. Yi, “Hash function based on chaotic tent maps,”
IEEE Trans. Circuits and Systems II, vol. 52, no. 6,
pp. 354-357, 2005.
[5] T. Habutsu, Y. Nishio, I. Sasase and S. Mori, “A secretkey cryptosystem using a chaotic map,” Trans. IEICE,
vol. E73, no. 7, pp. 1041-1044, 1990.
[6] M. Harada, Y. Nishio and A. Ushida, “A cryptosystemusing two chaotic maps,” Proceedings of NOLTA’99,
vol. 2, pp. 609-611, 1999.
[7] N. Masuda and K. Aihara, “Cryptosystems with discretized chaotic maps,” IEEE Trans. Circuits and Systems I, vol. 49, pp. 28-40, 2002.
[8] L. Kocarev and Z. Tasev, “Public-key encryption basedon Chebyshev maps, ” Proceedings of ISCAS’03, vol.
3, pp. 28-31, 2003.
[9] K. Y. Cheong and T. Koshiba, “More on security of
public-key cryptosystems based on Chebyshev polynomials,” IEEE Trans. Circuits and Systems II, vol. 54,
no. 9, pp. 795-799, 2007.
[10] “The GNU MP Bignum Library,” http://gmplib. org/

- 387 -