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Physics Letters A 376 (2012) 1003–1010

Contents lists available at SciVerse ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

An improved chaotic cryptosystem based on circular bit shift and XOR operations

Shu-Jiang Xu

a,b,c,∗

, Xiu-Bo Chen

a,b

, Ru Zhang

, Yi-Xian Yang

, Yu-Cui Guo

Information Security Center, Beijing University of Posts and Telecommunications, Beijing 100876, China

State Key Laboratory of Information Security (Graduate University of Chinese Academy of Sciences), Beijing 100049, China

Shandong Provincial Key Laboratory of Computer Network, Shandong Computer Science Center, Jinan 250014, China

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

article info abstract

Article history:

Received 4 October 2011

Received in revised form 19 December 2011

Accepted 24 January 2012

Available online 30 January 2012

Communicated by A.R. Bishop

Keywords:

Chaos

Chaotic cryptosystem

Image encryption

Chen’s chaotic system

A type of chaotic encryption scheme by combining circular bit shift with XOR operations was proposed

in 2006 based on iterating chaotic maps. Soon after the proposal, it was cryptanalyzed and improved.

Unfortunately, there are still two drawbacks in the two improved schemes. To strengthen the performance

of the focused type of scheme, a new improved scheme based on Chen’s chaotic system is proposed in

this Letter. Simulation results and theoretical analysis show that our improved scheme is immune to

information extracting by chosen plaintext attack and has expected cryptographic properties.

1. Introduction

With the rapid development of information technology and net-

work technology, digital images and other multimedia are more

commonly and frequently transmitted in public communication

network. Therefore, it is particularly important to protect the im-

age data against illegal purposes. So image encryption technol-

ogy becomes an important issue of cryptography. Image data have

strong correlations among adjacent pixels. Statistical analysis on

large amounts of images shows that averagely adjacent 8 to 16

pixels are correlative in horizontal, vertical, and diagonal directions

for both natural and computer-graphical images [1].Moreover,due

to some intrinsic features of images, such as bulk data capacity

and high redundancy, encryption of images is quite different from

that of texts [2]. As a result, conventional cipher algorithms, such

as DES, IDEA, etc., are not directly suitable for image encryption.

Therefore, many scholars have made an effort to investigate many

new image encryption schemes in order to promote communi-

cation security recently [1–16]. A novel digital image encryption

algorithm is presented by utilizing a new multiple-parameter dis-

crete fractional random transform [3]. In the year 2011, N. Zhou

et al. ﬁrstly introduced the fractional Mellin transform into the

ﬁeld of image security and proposed an originally novel nonlinear

image encryption scheme with large key space and corresponding

Corresponding author at: Information Security Center, Beijing University of Posts

and Telecommunications, Beijing 100876, China. Tel.: +86 10 62283086.

E-mail address: xushj@keylab.net (S.-J. Xu).

optoelectronic hybrid structure to overcome the drawbacks of im-

age encryption schemes based linear transforms, such as Fourier

transform [4]. Anil Kumar et al. propose an extended substitution-

diffusion based image cipher using chaotic standard map [12].By

spatial bit-level permutation and high-dimension chaotic system,

a color image encryption is proposed in [13].Amongthesealgo-

rithms, the chaos-based cryptography has given a new and eﬃcient

way to develop fast and secure image encryption algorithms.

Chaos, which has many good properties, such as ergodicity,

sensitive dependence on initial conditions and random-like behav-

iors, is an aperiodic complicated motion modality and nonlinear

phenomenon [17]. Most of the properties have granted chaotic

cryptosystems as a promising alternative for the conventional cryp-

tographic algorithms [1]. Matthews ﬁrst proposed the chaos-based

encryption scheme in 1989 [6], and Fridrich ﬁrst adopted chaotic

map into image encryption in 1997 [7]. Since then, many chaos-

based image encryption algorithms have been designed to realize

secure communications [1,8–16].

With a pseudorandom bit sequence generated by iterating

chaotic maps, a type of chaotic encryption scheme which was

based on circular bit shift and XOR operations was proposed in

[15,16]. In this type of algorithm, the plaintext block was per-

muted by a circular bit shift approach and then encrypted by the

XOR operation block by block. However, it was shown that there

are some defects with this type of algorithms in [18]:(1)Neither

the chaotic trajectories of the logistic map nor those of the de-

layed chaotic neural networks have a uniform distribution, which

leads to insuﬃcient randomness of the chaos-based pseudorandom

bit sequence generated from these chaotic trajectories. (2) Low

doi:10.1016/j.physleta.2012.01.040

1004 S.-J. Xu et al. / Physics Letters A 376 (2012) 1003–1010

sensitivity of encryption to plaintexts. (3) Not secure against the

differential known-plaintext attack and the chosen plaintext at-

tack. It was pointed out that the type of scheme facilitates leakage

of the information and is vulnerable to the chosen plaintext at-

tack [19]. To obtain higher security, an improved scheme was sug-

gested [19], but it was found that the ﬁrst several bits in quantiﬁed

sequence generated by chaotic sequence are still not sensitive to

the least signiﬁcant bits of chaos initial state and the improved

scheme cannot resist chosen plaintext attack [20].Basedontwo

chaotic maps, an improved scheme of [15,16] was proposed [21],

in which the plaintext is ﬁrst encrypted using the XOR operation

block by block, and then it is encrypted using the circular bit shift

operation block by block in the inverse order of the ﬁrst round

of encryption. Wang et al. pointed out that the encryption algo-

rithm presented in [15] is vulnerable to chosen plaintext attack

because of the plaintext-independent key stream, and suggested

a straightforward improved scheme utilizing ciphertext feedback

mechanism [22]. Nevertheless, Ben Farah et al. noticed that the

improved scheme [22] still was insecure against chosen plain-

text attack since the ﬁrst block of key stream is also plaintext-

independent, and proposed an improved algorithm by generating a

random stream as the ﬁrst block of key stream [23].Theimproved

scheme [22] was also analyzed using key stream attack and im-

proved based on block cryptosystem by Guo et al. [24].

Because the ﬁrst four steps of Guo’s improved scheme [24] is

the same as that of Wang’s scheme [22], the ﬁrst block of the key

stream in Guo’s improved algorithm is also plaintext-independent.

Therefore, the improved scheme [24] is unable to withstand cho-

sen plaintext attack according to [23].Fortheimprovedscheme

[23], each encrypting process will generate a random key stream

as the ﬁrst block of key, which means that the secret key changes

every time the algorithm is used. We think the improved scheme

[23] is debatable, since this type of encryption scheme is a block

cryptosystem rather than a stream cryptosystem or a one-time

pad scheme. If a random key block is generated for several time

encryption processes, the improved scheme [23] will be insecure

against chosen plaintext attack. Note that the random stream can

be used as part of the ciphertext instead of part of the secret

key. Furthermore, the pseudorandom bit sequence was generated

by the logistic map in these improved schemes [21–24],sothe

second defect shown in the above paragraph still exists in the

schemes. When a plaintext byte is changed, only the ciphertext

bytes behind the corresponding ciphertext byte are changed, so

the improved schemes [21–24] also have low sensitivity to small

changes of the plaintext. As a result, the two measures of differ-

ential analysis, NPCR and UACI, are still not reasonable in schemes

[21–24]. Following the main idea of this type of chaos-based en-

cryption scheme which combines the circular bit shift operations

and XOR operations, an improved scheme of it is proposed in this

Letter so as to strengthen its performance.

TherestofthisLetterisorganizedasfollows.Section2 presents

the approach to generate a statistical random number sequence by

iterating the Chen’s chaotic system (CCS). An improved chaotic im-

age encryption algorithm is introduced brieﬂy in Section 3 based

on the CCS. The experimental result is given in Section 4. Section 5

is devoted to security analysis. Finally, Section 6 concludes this Let-

ter.

2. Random numb er sequences generation

2.1. Chen’s chaotic system

One-dimensional chaotic system which has the advantages of

high-level eﬃciency and is implicit [25],suchasLogisticmap,is

being widely used now. But their weaknesses, such as small key

space and weak security, are also known to us obviously [26].

Fig. 1. 3D Chen’s chaotic system.

To overcome these drawbacks, a three-dimensional (3D) chaotic

system is adopted in this Letter. With more than one initial con-

ditions and control parameters, high-dimensional chaotic systems

are most complex and have a big key space which has useful effect

on the chaotic cryptosystem.

In 1999, Prof. G. Chen ﬁrst presented CCS [27]:



x = a(y − x),

y =(c −a)x − xz + cy,

z = xy −bz,

(1)

where a, b and c are parameters. Choosing a = 35, b = 3, c ∈

[

20, 28.4], the system enters chaotic domain which is shown in

Fig. 1. The equations of CCS are quite similar to those of Lorenz

system, while the topologically properties of the two systems are

not equivalent, essentially due to the parameter c in front of the

state variable y, which can lead to abundant dynamic characters

of CCS. Therefore, the dynamical property of CCS is more compli-

cated than Lorenz chaotic system. This feature is very useful in

secure communications [28].Eq.(1) will be solved by means of

the fourth-order Runge–Kutta method. The time step size h which

can be considered as one of the secret key here is chosen as 0.001.

2.2. Random number sequence

In this section, we will show that a statistical random sequence

can be generated by CCS. As a result, the ﬁrst defect shown in

Section 1 can be successfully avoided. Suppose that x is a double-

point value, and the random number is generated as follows:

mod



x ×10

−



x ×10



, 256



(2)

where mod(x, y) returns the remainder after division, x rounds

the elements of x to the nearest integers less than or equal to x.

Every time the CCS which has three variable values x

, y

and z

is iterated, three random numbers can be got. Iterate CCS contin-

uously, a random number sequence D

={D

| x = x

, y

, z

} can be

obtained.

For the convenience to test randomness of the generated se-

quence D, we can convert the decimal number D

to binary num-

ber and divide it into bits. The obtained binary sequence B

={B

= 0, 1} can be tested as below. Federal Information Processing

Standard 140-2 which was published by the National Institute of

Standards and Technology (NIST) deﬁnes the security requirements

that must be satisﬁed by a cryptographic module used to protect

unclassiﬁed information within information technology systems. In

the documentation of FIPS PUB 140-2, statistical random number

generator tests are deﬁned as follows [29]: A single bit stream of

20 000 consecutive bits of output from RNG shall be subjected to

monobit test, poker test, runs test and long runs test. The mono bit

test measures whether the number of 0s and 1s produced by the

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