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.

A Cryptosystem Based on Iterations of Chaotic Map
Shuichi AONO† and Yoshifumi NISHIO†† Dept. of Electrical and Electronic Eng., 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 a change in initial conditions and parameters, and a long-term forecast becomes impossible by the iterations of a chaotic map. These features look similar to the properties of thecryptology. For that reason, it is eﬀective to use chaotic maps for cryptosystems. In this research, we propose acryptosystem by using iterations of a chaotic map. This cryptosystem uses expansion map for encryption and decryption. This cryptosystem is a symmetric-key cryptography that has a public key. We investigate the vulnerabilityof this cryptosystem.
Key words chaotic cryptosystem, logistic map

1. Introduction

cently some researchers are proposed a public-key cryptography based on chaotic maps [9].

A chaotic map has various features. A chaotic map has

In this research, we propose a cryptosystem by using iter-

sensitivity to a change in initial conditions and parameters,

ations of a chaotic map. And we investigate the vulnerabil-

and a long-term forecast becomes impossible by the itera-

ity of this cryptosystem. This cryptosystem uses expansion

tions of a chaotic map. These features look similar to themap for encryption and decryption. The decryptor and theproperties of the cryptology. For that reason, it is eﬀective toencryptor have each private key, and they encrypts to cipher-use chaotic maps for cryptosystems. The chaotic cryptosys-

text and decrypts to plaintext with each private key and atem is researched as an application of chaos in an engineeringcommon private key. This cryptosystem is a symmetric-key

ﬁeld [1-4].

cryptography that has characteristic of a public-key cryptog-

A lot of chaotic cryptosystems that use the expansion andthe reduction of the chaotic map for the encryption and decryption are reported [5][6]. On the other hand, many re-raphy.

2. A Chaotic Map

searchers are reported about the disadvantages of cryptosys-

In this research, we use a modiﬁed logistic map. The mod-tem using the chaos map [7][8]. There is a possibility toiﬁed logistic map is one of the simplest chaotic maps. The

be linear attacked when the piecewise-linear map is used in

modiﬁed logistic map is expressed as the following equation:

map is demonstrated when the direction of the expansion isXk2
 Xk+1 = α Xk (1 − 2α )used for the encryption. It is undesirable to use reduced map

the cryptosystem. In addition, the advantages of the chaoticfor cryptosystem.

(0 <= Xk <= α)(1)+1−2α+1−2α
Xk+1 = ( Xk1−α)(2 − Xk1−α)(α < Xk <= 1)

Moreover, a symmetric-key cryptography has the problem

Here, α is a parameter changing the top of the map. Thisof the delivery of the key distribution. This problem can bemap shown in Fig. 1. The generated sequences looks likesolved by using a public-key cryptography. However, there

the uniform random number. Figure 2 shows an example

are two problems in the application of the chaotic map toof chaotic sequences generated by the modiﬁed logistic map.

the public-key cryptography. One is that the trap door is

The behavior like the uniform random number is seen innecessary to apply the public key cryptosystem. It is diﬃ-α = 4.0.

cult to develop trap door in the chaotic maps. The other isThis map has sensitivity to a change in parameter α.

that the equation form is opened to the public. A chaotic

Therefore, a long-term forecast becomes impossible by themap is shown by the equation based on determinism. If theiterations of a map. However, the modiﬁed logistic map isequation of the chaotic map and their parameters are opened

based on an easy equation. Even if eavesdroppers does notto the public, the behavior of the sequences is known. Re-know the value of α, parameter α can be obtained from the—1—

Fig. 1 A modiﬁed logistic map F .

Fig. 3 Block diagram of the cryptosystem.

Fig. 2 Chaotic sequence generated by the modiﬁed logistic map.

•α : common private key.

•A : private key.

3. 2 Encryption Process
We explain a encryption process of this cryptosystem. The

relation of two successive numbers. In other words, it isencryptor chooses the value B as a number of iterations,necessary to know the number of iterations of the modiﬁed

where B is an arbitrary value. Encryptor encrypts by using

logistic map to obtain α. If the number of iterations k is ana private key B.

unknowns, it is very diﬃcult to obtain correct α from theThe encryption functions are described as follows :value of X0 and Xk . We propose the chaotic cryptosystem
C1 = M + F B (XA ) = M + XA+B

using this feature.

C2 = F B (X0 ) = XB

(2)

3. Cryptosystemhere, M is a plaintext.
We propose a cryptosystem by using iteration of a modiﬁed

Send this value (C1 , C2 ) as a ciphertext to the receiver.

logistic map. The simpliﬁed block diagram of the cryptosys-•

C1 , C2 : ciphertexts.

tem is shown in Fig. 3. This cryptosysytem is composed of•B : private key.

the following three parts. Key generation, encryption process and decryption process.

3. 3 Decryption Process
In the decryption process, decryptor calculates C1 −A

3. 1 Key Generation
This cryptosystem uses three kind of keys, a public key, aF (C2 ) by using a private key A.
C1 − F A (C2 ) = M + XA+B − F A (XB )private key and a common private key. A common private= M + XA+B − XB+Akey means shared secret key between the decryptor and the=Mencryptor. And, the keys are decided by the decryptor. The
process of key generation is as follows.
The decryptor set an initial point X0 and a parameter α,
here α is a common private key. In addition, set a number of(3)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 common private key α and private key A fordecryption.

iterations A. This value A is a private key of the decryptor.
The decryptor keep a private key A to himself.

4. Requirements for Cryptosystem

ACalculate XA = F (X0 ), namely A-time iterations of a
modiﬁed logistic map F .
The decryptor can obtain X0 , XA as a public keys, α as acommon private key and A as a private key.
•X0 , XA : public keys.

4. 1 Key and Plaintext Sizes
The key space size and plaintext size are required over 128
bits (40 digits) for the defense against brute force attack. In
this cryptosystem, a public keys, a common key and plaintext size are deﬁned as follows :—2—•

α 40 digits.

•X0 , XA (∈ 0, 1) and M 40 digits.

sizes are deﬁned the same as the plaintext.
•C1 and C2 40 digits.

4. 2 Range of Parameter α

4. 4 Number of Iterations

Next, we investigate the range of parameter α. There is a

We investigate the number of iterations as a private keys

possibility that the distribution of the generated sequences

A and B. A and B are arbitrary value. The cipher-breaking

has bias for some chosen value of the key and chosen initial

becomes diﬃcult by increasing this value. However, if thisvalue. Figure 4 and 5 shows the distribution of the value ofvalue becomes longer, The computation time increases.

the XN in changing the α the case of N = 10 and the caseof N = 100.

In order to determine the number of iterations, we simulated the sequences with slight diﬀerent keys. Figure 6 shows
the diﬀerence of the value between the case of α = 0.49 andthe case of α = 0.49 + 10−40 . Horizontal axis shows thenumber of the iterations N , vertical axis shows the diﬀerence between the generated sequences.

Fig. 4 The distribution of the XN the case of N = 10 .

Fig. 6 The diﬀerence of two value.

From this ﬁgure, we can see the diﬀerence of two valuesthat have changed nonuniformity in the case of N > 150.
There are not so many diﬀerences of two values until the iteration exceeds 150 times. It is undesirable to use the keyfor cryptosystem. Therefore we determine that the numberof iterations are arbitrary value over 150 times.
Fig. 5 The distribution of the XN the case of N = 100 .

•A and B > 150 times.

5. Security of The Propose Cryptosystem
From Fig. 4, we can see that the bias is caused at thevalue of α close to 0 and 1. This bias decreases according toincreasing the iterations of map. If the generated sequences
has some bias, the eavesdroppers does not calculate correct
keys. Because the decryptor can set an arbitrary the common private key α and arbitrary the private key A. In thisresearch, we deﬁned α(∈ 0.1, 0.9).
•5. 1 A Public Keys
This cryptosystem uses the public-key cryptography for apart of the cryptosystem, and the number of the delivery of
the keys are decreased. It is very diﬃcult for eavesdroppers
to obtain correct α and A from the value of X0 and Xk . So,
we open the X0 and the XA to the public as a public keys.
We explain that reason giving examples. When the publicα(∈ 0.1, 0.9) 40 digits.

keys are X0 = 0.41234 and XA = 0.24739, the eavesdroppers

4. 3 Ciphertext Size

are attacked by the brute force attack. Table 1 shows the re-In this cryptosystem, we do not expanse and reduce thesults of the brute force attack for the limited case of N < 50

value of the plaintext. We have to think only about the eﬀect

and α = 5 digits.

of rounding error of the sequences generated by a modiﬁed

This results are solutions in limited case, however eaves-logistic map. And so, we truncate a number to 40 decimal

dropper can detect a lots of a values. An important thing

places by the iteration of the map. Therefore, the ciphertext—3—

Table 1 The results of the brute force attack.
Private key α Iterations A Private key α Iterations A

against the CPA. When this cryptosystem is attacked by aCPA, and encryptor selects the same value B, the eavesdrop-0.04439

190.51144

80.04719

440.51929

450.06094

150.52547

250.10328

20.54957

19

this cryptosystem. However, it is diﬃcult for eavesdroppers

0.13988

210.55229

27

to obtain correct A, B and α from the value of X0 , XA ,0.15702170.62369

9

XB and XA+B . Even if a discrete-valued of the chaotic se-0.15839

240.64520

12

quences are known to the eavesdroppers, the eavesdroppers

0.18736460.64959

35

have to detect two successive numbers or the number of the0.19175

400.67215

240.24723

210.74644

460.30044470.74863

220.33086

30.79956

45of the iterations B at the every encryption, also same for the0.37352

350.82766

34

decryptor. The security of the cryptosytem becomes more

0.43097

170.83572

38

safety by setting the diﬀerent key at the encryption.

0.44703390.85982

340.45617

70.88199

490.47851

230.95105

230.48088

270.97876

380.50042

36

pers can easily obtain two values as XA+B and XB . Because,the value of the plaintext is not changed and expanded it initerations to calculate the parameter α.
In addition, the encryptor can easily change the number

6. Conclusions
In this research, we have proposed a cryptosystem by using
iterations of a modiﬁed chaotic map. And we have investigated vulnerability of this cryptosystem. The proposed cryptosystem is a symmetric-key cryptography that looks like ais that the all of the values are correct keys for the eaves-

public-key cryptography that has both of a private key anddropper who knows only X0 = 0.41234 and XA = 0.24739.

a common private key. The number of the delivery of theBut a really correct private keys are only one in their val-keys has decreased to using this cryptosystem.

ues. When the eavesdroppers chooses a really correct key inAs the future subject, we would like to investigate security

this Table 1, it is necessary to know the value of the α orof the cryptosystem in more detail. And we apply the cryp-

the value of the A. Only the decryptor and the encryptor

tosytem by using iterations of a chaotic map to the systemknows that parameters. Moreover, there is a possibility thatof the personal authentication.

all of the parameter α generates the value of the XA after
the N -times iterations. Because the generated sequences by
the modiﬁed logistic map are a long-term sequences. It isdiﬃcult to establish the value of the X0 does not becomesthe value of the XA by repeating N times of the map witha certain α.
For that reason, we thought that the value of α and thevalue of A did not calculate even if the value of the X0 andthe value of the XA were opened to the public. And we used
X0 and XA as a public keys.
We can think about the ciphertexts C1 and C2 in a similar
way the above. In actuality, the sequences generated by themodiﬁed logistic map has period. If the generated sequences
becomes true random number, here is a possibility that all of
the parameter α that shown in Table 1 generates the valueof the XB as a C2 after the N -time iterations.
5. 2 A 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.
This propose cryptosystem is a simple but eﬀective system

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