社団法人 電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS

信学技報
TECHNICAL REPORT OF IEICE.

Chaotic Map with Chaotically Changing Parameter for a Cryptosystem
Shuichi AONO† and Yoshifumi NISHIO†† Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: †{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. These features look similar to the features of cryptology.
For this reason, it is eﬀective to use chaotic maps for cryptosystems. In this research, we propose a two-dimensional
chaotic map with a chaotically changing parameter for a cryptosystem. And we propose a cryptosystem using atwo-dimensional chaotic map. A characteristic of the proposed cryptosystem is that a diﬀerent ciphertext is generated from the same plaintext. The ciphertext becomes the value that depends on an encryptor. This characteristic
is one of the important features in public-key cryptography. This cryptosystem is symmetric-key cryptography thathas a characteristic of public-key cryptography. We investigate the vulnerability of this cryptosystem.
Key words chaotic map, cryptosystem

1. Introduction

An encryptor select an arbitrary value as a private key. The
plaintext becomes the ciphertext that depends on this private

A chaotic map has sensitivity to a change in the initial con-key. We consider that this feature is the interesting feature

ditions and parameters, and a long-term forecast becomes

for the development of the chaotic cryptosystem. Because,

impossible by iterating a chaotic map. These features lookthis feature means that the slightly diﬀerent condition has tosimilar to the features of cryptology. For this reason, it ischange in a randomly diﬀerent. It may be no exaggerationeﬀective to use chaotic maps for cryptosystems. A chaotic

to say that this is a characteristic of the chaotic map.

cryptosystem is researched for the application of chaos inengineering ﬁeld [1-4].

In this research, we propose a two-dimensional chaotic map
with a chaotically changing parameter for a cryptosystem.

Many chaotic cryptosystems that use the expansion andThis chaotic map has a changing parameter generated by

the reduction of chaotic maps for encryption and decryption

the logistic map. And we propose a cryptosystem using thisare reported [5][6]. Figure 1 shows the basic chaotic cryp-

two-dimensional chaotic map. A characteristic of the pro-tosystem. If we use one-to-n map as the expansion map, theposed cryptosystem is that a diﬀerent ciphertext is gener-

plaintext cannot be uniquely decrypted, because calculating

ated from the same plaintext. The ciphertext becomes thethe reverse-map becomes n-to-one map. Namely, it is neces-value that depends on an encryptor. This characteristic issary to propose the cryptosystem that one-to-one mapping

one of the important features in public-key cryptography.

is realized between the plaintext space and the ciphertext

This cryptosystem is symmetric-key cryptography that has

space in the decryption [7]. By using only an expansion map

a characteristic of public-key cryptography. We investigate

for encryption and decryption, one-to-one mapping is real-

the vulnerability of this cryptosystem.

ized 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 thekey distribution can be solved using public-key cryptography. Though this is an important advantage of public-keycryptography, public-key cryptography has other advantage.

Fig. 1 Chaotic cryptosystem

A diﬀerent ciphertext is generated from the same plaintext.
—1—2. Two-dimensional Chaotic Map
In this research, we use an expansion map for encryption
and decryption. By using only an expansion map, even if
reverse-map turns into n-to-one map, it is possible to decrypta correct plaintext. Therefore, we can use a more complex
chaotic map that the calculation of the reverse-map becomesdiﬃcult for eavesdroppers.
In order to make calculation of the reverse-map diﬃcult,
we propose a two-dimensional chaotic map. The proposed
two-dimensional chaotic map is expressed as the following
equation:

Xn2

Xn (1 −)
Xn+1 =Y2YnnF :(a)(0 <= Xn <= Yn )Xn + 1 − 2Yn
Xn + 1 − 2YnXn+1 = ()(2 −)
1−Y
1 − Ynn(1)(Yn < Xn <= 1)

where


Yn2)
Yn+1 = Yn (1 −


α2α


(0 <= Yn <= α)

Yn + 1 − 2α
Yn + 1 − 2αYn+1 = ()(2 −)
1−α1−α(α < Y < 1)(2)n =This chaotic map is changed in the parameter α by chaotic(b)sequences generated by the logistic map. We consider thatthe cipher-breaking and long-term forecast become more dif-Fig. 2 Chaotic maps (α = 0.5) (a) Xn , (b) Yn

ﬁcult by changing a parameter α in chaos. This maps areshown in Fig. 2. We can see that the chaotic attractor becomes more complex as compared to the logistic map. Figure
3 shows the chaotic sequences of Xn . This map has sensitivity to a change in parameters α and initial states. Therefore, a long-term forecast becomes impossible by iterating achaotic map.
However, the proposed two-dimensional chaotic map is expressed by simple equations. If the number of iterations nand the parameter α are known, it is easy to calculate Xnand Yn from initial values X0 and Y0 . In other words, it isnecessary to know both the number of iterations and param-Fig. 3 Chaotic sequences generated by proposed chaotic map

eter α to obtain the correct Xn and Yn . If the number ofiterations n or parameter α are unknown, it is diﬃcult tocalculate Xn and Yn .

3. Proposed Cryptosystem

the decryption process. We use the following characteristics
of a chaos map to solve this problem.
F A ◦ F B (X0 ) = F B ◦ F A (X0 )(3)

We propose a cryptosystem by using iteration of a twodimensional chaotic map. This cryptosystem is composed of

3. 1 Key generation

the following three parts. The key generation, the encryption

This cryptosystem uses three kind of keys, a public key,process and the decryption process. In order to combine thea private key and a shared secret key. A shared secret key

concept of the private key and symmetric-key cryptography,

means private key between the decryptor and the encryptor.

the chosen private key must be negated by the decryptor inAnd, the keys are decided by the decryptor. The process of—2—

is performed as follows.

key generation is as follows.
A decryptor sets an initial points for X0 , Y0 and a param-（ 1 ） Divide the interval [0, 1] into l class intervals.

eter α. Here, α is a shared secret key. In addition, set the（ 2 ） Calculate Xn with the diﬀerent key α. Here, X0 isnumber of iterations A. This value A is a private key fora ﬁxed value. Make an l × l contingency table fi .

the decryptor. The decryptor keep a private key A to him-（ 3 ） Calculate

self. (XA , YA ) = F A (X0 , Y0 ), namely, A-time iterations ofχ2 =the two-dimensional chaotic map F are calculated.
The decryptor can obtain X0 , XA , Y0 , YA as public keys,l2∑(fi − f ′ )ii=1(6)fi′

where fi′ is an ideal value. However, the two-dimensional

α as a shared secret key and A as a private key.
•X0 , XA , Y0 , YA : public key.

chaotic map does not pass the χw test. Because this map•

α : shared secret key.

has some bias in the distribution of the generated sequence.

•A : private key.

In this research, we use the frequency of Xn with the random
α as ideal.

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 byThe upper 5% of this χ2 test is 16.9. If the value of χ2
test is 16.9 or less, the uniformity of the generated sequenceis guaranteed. The simulated results for 10 diﬀerent keys
and the number of iterations are shown in Table 1. Fromusing a private key B.

this table, we conﬁrm that all the distributions of Xn are

The encryption functions are described as follows :uniform. In other words, we can use any parameter α as
C1 = M + F B (XA ) = M + XA+B
C2 = F B (X0 ) = XBa shared secret key because the distribution of Xn is inde(4)C3 = F B (Y0 ) = YB

pendent on the parameter α of the two-dimensional chaoticmap. We determined that the range of the shared secret key
is α ∈ [0, 1].

where M is a plaintext.
(C1 , C2 , C3 ) are calculated. These values are sent to a receiver as ciphertexts.

Table 1 χ2 test for uniformity
Range of α N = 50 N = 100 N = 150•C1 , C2 , C3 : ciphertext.

•0 ∼ 0.1

B : private key.

0.1 ∼ 0.27.95

8.72

16.25

0.2 ∼ 0.3

10.17

16.01

13.11

0.3 ∼ 0.4

14.03

6.45

9.51

0.4 ∼ 0.5

9.83

14.70

14.68

0.5 ∼ 0.6

7.38

9.78

5.46

0.6 ∼ 0.7

11.46

14.56

15.91

0.7 ∼ 0.8

10.76

8.31

10.86

0.8 ∼ 0.9

16.90

11.53

13.27

0.9 ∼ 1.0

13.28

15.87

3.74

3. 3 Decryption process
In the decryption process, a decryptor calculates C1 −F A (C2 ) by using C3 and a private key A.
C1 − F (C2 ) = M + XA+B − F (XB )A

A= M + XA+B − XB+A

(5)

=M

16.75

16.77

8.03

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 α and a private key A for
decryption.

4. Security Analyses

Next, we investigate numbers of iterations A and B.
Cipher-breaking becomes diﬃcult by increasing these parameters. The calculation time of the proposed cryptosystemdepends on the number of iterations. In order to determine
the number of iterations, we simulated the sequences with

4. 1 Brute force attackThe key space size α 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. If there is no eﬀective shortcut attack, thekey space size with 128 bits is computationally-secure key
space.
We consider the range of the parameter α. We investigate
the correlation of α and Xn . We use χ2 test for the uniformity to determine the range of α. χ2 test for the uniformity

slightly diﬀerent keys. Figure 4 shows the diﬀerence of thegenerated value Xn between the case of α = 0.51 and thecase of α = 0.51 + 10−40 . The horizontal axis shows thenumber of iterations, and the vertical axis shows the diﬀerence between the generated sequences.
From this ﬁgure, there are no large diﬀerences between
the two values unless the iteration exceeds 150 times. It isundesirable to use the range of N < 150 as keys. We cansee the diﬀerence between the two values that have changed—3—

nonuniformity for N > 150. Therefore, we determine thateavesdroppers, the eavesdroppers have to detect two succes-

the number of iterations is arbitrary but exceeds 150.

sive numbers or the number of iterations to calculate theparameter α.
In addition, the encryptor can easily change the number ofiterations B at the every encryption. The encryptor can generate a diﬀerent ciphertext from the same plaintext. This feature is an advantage compared with the conventional chaotic
symmetric-key cryptosystem. The security of the cryptosystem becomes more safety by setting the diﬀerent key at theencryption.

5. Conclusions
In this research, we have proposed a two-dimensional
Fig. 4 Diﬀerence of two values

chaotic map with a chaotically changing parameter. And we
propose a cryptosystem using the two-dimensional chaotic

Next, we consider the rounding error by the discrete

map. A characteristic of the proposed cryptosystem is thatchaotic map. We truncate Xn and Yn to 40 decimal places

a diﬀerent ciphertext is generated from the same plaintext.

by iterating of the map. Therefore, the chaos space is cer-We have investigated vulnerability of this cryptosystem.

tainly reduced. However, the reduction of the chaos space

As the future subject, the security of the proposed cryp-does not aﬀect the key space. We secure enough key spacetosystem will be investigated in more detail. We will develop

in consideration of rounding error.

the secret sharing scheme by using iterations of a chaotic

The proposed cryptosystem uses the public-key cryptog-map.

raphy for a part of the cryptosystem, and the number of theReferencesdelivery of keys are decreased. There is a possibility that

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all of the parameter α generates the value of XA after N times iterations. Because the generated sequences by thetwo-dimensional chaotic map are a long-term sequences andthere are no bias for the range of parameter. For that reason, It is diﬃcult for eavesdroppers to obtain correct α andA from the value of X0 , Y0 , XA and YA . We thought thatthe value of α and the value of A did not be calculated even
if the value of X0 and the value of XA were opened to thepublic.
4. 2 Chosen plaintext attack
Next, we consider a chosen plaintext attack (CPA) for thiscryptosystem. 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 systemagainst the CPA. If the proposed cryptosystem is attacked
by the CPA, and the encryptor selects the same value B,
the eavesdroppers can easily obtain two values XA+B , YB
and XB . Because, the proposed cryptosystem is considered
to be sort of stream cipher. Namely the value of the plaintext is not changed and expanded it in this cryptosystem.
However, it is diﬃcult for eavesdroppers to obtain correct A,
B and α from the public keys and the ciphertexts. Even ifa discrete-valued of the chaotic sequences are known to the—4—