Chinese Character Encryption Algorithm Based on Logistic Mapping

Wenping Guo, Ying chen, Xiaoming Zhao School of Mathematics and Information Engineering, Taizhou University

Linhai, China guo _ wp@126.com

Abstract-Chaos system is a kind of complicated, nonlinear dynamic system. The complexity, pseudo-random and sensitivity to initial conditions of chaotic sequence have close relationship to cryptographic properties. We analyze the typical dynamic system-logistic mapping. On the basis of this analysis, we present a novel Chinese character encryption algorithm based on logistic mapping. This new methods computes simply and rapidly with high security. The results are described in the end.

Keywords- Chinese character; chaos; chaotic sequence; logistic mapping;

I. INTRODUCTION

With the rapid development of Internet in china, security is becoming an important issue in the storage and communication of Chinese character. In many secure fields, such as public safety and military department, Chinese characters are required to be encrypted. Technologies of cryptography are the core of information security. Claude Shannon proved that the one-time pad cipher is unbreakable [1]. But most ciphers, apart from the one-time pad, can be broken with enough computational effort by brute force attack. So the information security is facing challenge. So in recent years, chaotic encryption has become a new research field.

Mathematically, chaos refers to a very specific kind of unpredictability: deterministic behavior that is very sensitive to its initial conditions [2]. In other words, infinitesimal variations in initial conditions for a chaotic dynamic system lead to large variations in behavior. Chaotic systems consequently appear disordered and random. However, they are actually deterministic systems governed by physical or mathematical laws, and so are completely predictable given perfect knowledge of the initial conditions. In other words, a chaotic system will always exhibit the same behavior when seeded with the same initial conditions - there is no inherent randomness in this regard [3].

Edward Lorenz and Henri Poincare were early pioneers of chaos theory, and James Gleick's 1987 book Chaos: Making a New Science helped to popularize the field. More recently, computer scientist Christopher Langton in 1990 coined the phrase "edge of chaos" to refer to the behavior of certain classes of cellular automata [4].

Chaotic system is a kind of complicated, nonlinear dynamic system. It has perfect security with the following propertied: good pseudo-random, orbital inscrutability,

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extreme sensitivity about the initial value and the control parameter, etc. In this paper, we present a novel Chinese character encryption algorithm based on logistic mapping.

II. CHAOTIC SYSTEM AND LOGISTIC MAPPING

A. Definition of Chaotic system

Chaos theory is a field of study in mathematics, physics, and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general [5]. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved [6]. In other words, the deterministic nature of these systems does not make them predictable [7]. This behavior is known as deterministic chaos, or simply chaos.

Although there is no universally accepted mathematical definition of chaos, the first mathematical definition of chaos was introduced by Li and Yorke, and it became one of the most discussed topics for the last several decades. Li and Yorke proved that if a map on an interval has a point of period three, then it has points of all periods. Period three implies chaos. See THEOREM lr81.

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d S a < b < c (or d � II > b > C',. Tilt''' TI: lor t'�" .I. • 1.2.'" tltt'" iJ II JHriodk PO;If' ill J had,., ptriod I:. FUf1htrmou. 1"2: /It," is 1111 IlttCOlifI/ab/1 HI S CJ (colttainillK 110 ptriodic poi",s). "'hid saliS;lJ

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(2.2)

(8) For «try pes Qnd ptriod� poirtl q e J.

Ii,!,s�p I ,... .. (p) - F"'(q)1 > O.

B. Typical Chaotic System and Logistic Mapping

A commonly-used definition says that chaos is an aperiodic, non-convergent process, which sensitively depends on the initial state. It is determined but stochastic­like and exists in the nonlinear dynamic system. A discrete temporal nonlinear dynamic system can be described as following formula (1) [9]:

(1)

Where Xk E p, k=O, I , 2, 3 ...... , we call these states.

While T: p --7 P is a mapping, which links the current

state Xk to the next state Xk + I . So if we start with the initial state xo , and do mapping T repeatedly, we will get a sequence {Xk; k=O, I , 2, 3 ...... }. This sequence is the track of the discrete system.

One simple but widely used dynamic system is a logistic mapping. It is defined as the following formula (2) [10]:

Xk + I = ,un(1-Xk) (2)

Where 0:::; f.1:::; 4 is branch parameter and Xk E (0,1). Research on Chaos dynamic system show that when

3.5699456:::; f.1:::; 4 , logistic mapping will be in chaos

condition. That is, with the initial state xo , the state sequence { Xk ; k=0, I, 2, 3 ...... }generated by logistic mapping is an aperiodic, nonconvergent course, which sensitively depends on the initial state. Fig.I shows the simulation results of logistic mapping while xo = 0.5 and iteration number = 300; From the Fig.!, we can see that the

range of j.l is from 0 to 4. The j.l closer it approaches 4, the numeric area of X is more even distributed from 0 to 1.

So the numeric of j.l should be more close to 4.

XO - 0.5; Iteration Nwnber - 300;

Figure I. Bifurcation diagram of logistic mapping

When we determine the numeric of j.l, for example, f.1 =3.99, let's check the effect of the initial value to the

system. Fig.2 shows the D-value of two logistic mapping while initial state: xO=0.663489000 and xO=O.66348900I, f.1 =3.99. From the Fig.2, at the beginning of Iteration, the

D-value is very small, approximately equal to O. The D-value of two sequences shows irregular situation with the increase of Iteration. Small differences in initial conditions yield widely diverging outcomes for logistic mapping, rendering long-term prediction impossible in general. This is butterfly effect.

479

I

1\

M �r 0 jJ / '\ V I j �

I

Figure 2. Sensitivity of Xk on initial condition xo

III. CHINESE CHARACTER ENCRYPTION ALGORITHM

BASED ON LOGISTIC MAPPING

A Chinese character, also known as a Han character, is a logogram used in writing Chinese (hanzi). Chinese characters represent the oldest continuously used system of writing in the world. The number of Chinese characters contained in the Kangxi dictionary is approximately 47,035, although a large number of these are rarely used variants accumulated throughout history. Studies carried out in China have shown that literacy in the Chinese language requires a knowledge of only between two and three thousand characters.

Firstly, we create a Chinese character library. For simplicity, we select 2048 Chinese characters which used frequently in modem society of china.

Secondly, we establish the storage location for these 2048 Chinese characters. For example, the storage location of Chinese character "body" is number "657". The encryption and decryption operation are conducted in Chinese character library.

The Chinese character "me" is given as an example to illustrate the Chinese character encryption algorithm based on logistic mapping:

• Step 1: Input the encryption Chinese character

"me", f.1 and iteration number n , f.1 > 3.6. • Step 2: Query the storage location of "me" in

Chinese character library, get "b". • Step 3: XI = b / 2048, because XE [0, 1]. • Step 4: Iterative operation using logistic mapping

Xk + I = ,un(1-Xk) • Step 5: When iteration ends, get" Xn ".

• Step 6: Xn * 2048 = X * , we should get [x*] because X * may have fractional parts.

• Step 7: Query the Chinese character using [x*] in

the library, get the encrypted Chinese character. Program flow chart is shown in Fig.3.

Query the Chinese character using [x* 1

Figure 3. Program flow chart of encryption process

IV. EXPERIMENT RESULTS AND DISCUSSION

A. The process of encryption

:J!drJ��:ljiJt =OOXXJ: are the original characters. We input the original characters into the encryption program and

get the encrypted information while Ji = 3.99 and

n = 9000 , the results are shown in Fig.4. The encrypted characters are totally irrelevant.

305

194

1178

1583

49

884

1492

1 165

Figure 4. Encrypted characters

1226

509

480

B. The process of decryption

The process of decryption is reverse process of encryption. When we input the encrypted characters to the decryption program, we get the original characters, the results are shown in Fig.5.

305

194

1178

1583

49

884

Figure 5. Original characters

C. Sensitivity of initial condition

1492

1165

1226

509

When we change the variable Ji, set Ji = 3.88, the

encrypted characters are difference while we encryption the same original characters, the results are shown in Fig.6.

729

1482

1637

1229

558

84

386

1247

Figure 6. Sensitivity of initial condition

641

10

When we change the variable n, the results are also different as well. The sensitivity of initial value determines the encrypted characters are totally irrelevant. So we can get pleasant encryption effects.

D. The performance of security

The probability density function of logistic mapping is shown as follows formula (3):

�X)+� O<X<I} else

(3)

Set xo = 0.2 , Ji = 3.9999 and n = 30000 , we

analyzed the results by statistical methods. The distribution of logistic mapping is shown in table 1. We can see the logistic mapping have good random properties. This means the system has high security performance.

TABLE I. DISTRIBUTION OF LOGISTIC MAPPING

Distribution interval Number Percent

0 - 0.1 5919 19.73%

0.1 - 0.2 2685 8.95%

0.2 - 0.3 2218 7.39%

0.3 - 0.4 2140 7.13%

0.4 - 0.5 1890 6.30%

0.5 - 0.6 1973 6.58%

0.6 - 0.7 1937 6.46%

0.7 - 0.8 2310 7.70%

0.8 - 0.9 2733 9.11%

0.9 - I 6195 20.65%

v. CONCLUSION

With the rapid development of Internet in china, security is becoming an important issue in the storage and communication of Chinese character. Chaotic system is a kind of complicated, nonlinear dynamic system. The complexity, pseudo-random and sensitivity to initial conditions of chaotic sequence have close relationship to cryptographic properties. We analyze the typical dynamic system-logistic mapping. On the basis of this analysis, we

481

present a novel Chinese character encryption algorithm based on logistic mapping. The results indicate that the new method computes simply and rapidly with high security and can be widely applied to Chinese character encryption.

REFERENCES

[I] Claude E Shannon, Warren Weaver, "The Mathematical Theory of Communication", University of Illinois Press, 1999

[2] Saber N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, 1999, pp. 117.

[3] Wemdl, Charlotte, "What are the New Implications of Chaos for Unpredictability?". The British Journal for the Philosophy of Science 60,2009,pp. 195-220.

[4] Christopher G. Langton. "Computation at the edge of chaos". Physica D, vo1.42, 1990.

[5] Stephen H. Kellett, "In the Wake of Chaos: Unpredictable Order in Dynamical Systems", University of Chicago Press, 1993, pp. 32.

[6] Ralph H. Abraham and Yoshisuke Ueda (Ed.), "The Chaos Avant­Garde: Memoirs of the Early Days of Chaos Theory", World Scientific Publishing Company, 2001, pp. 232.

[7] Michael Barnsley, "Fractals Everywhere", Academic Press 1988, pp. 394.

[8] T.Y. Li, J.A.Yorke, "Period three implies chaos", American Mathematical Monthly, Vol. 82, 1975, pp.985-992.

[9] Hui Xiang, Digital Watermarking Systems with Chaotic Sequences. Security and Watermarking of Multimedia Contents, Vol.l,1999, pp.449-457.

[10] R. May, "Simple mathematical models with very complicated dynamics", Nature Vo1.261, 1976, pp:459-467