A bit-level image encryption algorithm based on spatiotemporal chaotic system and self-adaptive

A bit-level image encryption algorithm based on spatiotemporal chaotic system and self-adaptive

Lin Teng,Xingyuan Wang n

Faculty of Electronic Information and Electrical Engineering,Dalian University of Technology,Dalian116024,China

a r t i c l e i n f o

Article history:

Received5April2012

Received in revised form

3June2012

Accepted4June2012

Available online27June2012

Keywords:

Chaos encryption

Bit-level encryption

Self-adaptive

a b s t r a c t

This paper proposes a bit-level image encryption algorithm based on spatiotemporal chaotic system

which is self-adaptive.We use a bit-level encryption scheme to reduce the volume of data during

encryption and decryption in order to reduce the execution time.We also use the adaptive encryption

scheme to make the ciphered image dependent on the plain image to improve performance.Simulation

results show that the performance and security of the proposed encryption algorithm can encrypt

plaintext effectively and resist various typical attacks.

&2012Elsevier B.V.All rights reserved.

1.Introduction

With the rapid development of computer network technology,

a lot of sensitive information is transmitted over the network;

information security becomes more and more important.Image

information transmission has increased rapidly and image

encryption technology has drawn more attention.Nowadays,

image encryption schemes include two processes:substitution

and diffusion[1].The substitution stage permutes all the pixels as

a whole,without changing their values.In the diffusion stage,the

pixel values are modi?ed sequentially so that a tiny change in a

pixel spreads to as many pixels in the cipher-image as possible.

The two processes can achieve a satisfactory level of security.

Image encryption is different from text encryption due to some

inherent features such as bulk data capacity and high correlation

among pixels.Therefore,traditional cryptographic techniques such as

DES,IDES and RSA are no longer suitable for image encryption.The

properties of chaotic maps such as sensitivity to initial conditions and

random-like behavior have attracted attention to develop image

encryption algorithms.

In recent years,some chaos-based image encryption algorithms

have been developed[2–5].Xiang et al.[6]proposed a selective

image encryption using a spatiotemporal chaotic system.Liao et al.

[7]proposed a novel image encryption algorithm based on self-

adaptive wave transmission.In Refs.[8–10],CML-based spatiotem-

poral chaos systems were used in image encryption.

In this paper,we use a bit-level encryption scheme to reduce

the volume of data during encryption and decryption in order to

reduce the execution time.We also use the adaptive encryption

scheme to make the ciphered image dependent on the plain

image to improve performance,so the algorithm proposed in this

paper can resist various typical attacks.

The rest of this paper is organized as follows.Section2brie?y

reviews preliminary concepts in bit-level imaging,self-adaptive

encryption and coupled map lattice.Section3describes the

proposed encryption algorithm.The security of the scheme is

evaluated in Section4.Section5concludes the paper.

2.Preliminary

2.1.Bit-level image

In a gray-level image,the brightness between black and white

is quantized into an integer number of levels ranging from0to

255.The value of the pixel at coordinate(x,y)is denoted as f(x,y).

Each pixel can transform to an8bit binary value,given by

fex,yT?pe8Tpe7Tpe6Tpe5Tpe4Tpe3Tpe2Tpe1Te1T

So,the image can be divided into8binary images according to

the bit locations within a pixel.Select the sample image Lena of

size256?256and with256Gy levels(Fig.1),divide it into

8binary images(Fig.2)according to the bit locations within a

pixel.In Fig.2,Pic i represents the binary image obtained by

collecting the i th bits of all the plain-image pixels.

A bit can contain different amounts of information depending

on its position in the pixel.For example,‘‘1’’at the8th bit of a

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journal homepage:https://www.360docs.net/doc/744635279.html,/locate/optcom

Optics Communications

0030-4018/$-see front matter&2012Elsevier B.V.All rights reserved.

https://www.360docs.net/doc/744635279.html,/10.1016/j.optcom.2012.06.004

n Corresponding author.

E-mail addresses:tenglin@https://www.360docs.net/doc/744635279.html,(L.Teng),

wangxy@https://www.360docs.net/doc/744635279.html,(X.Wang).

Optics Communications285(2012)4048–4054

pixel represents 128(27),but it only represents 1(21)at the ?rst bit.The higher 4bits (8th,7th,6th and 5th)carry 94.125%of the total information of the image.On the other hand,the lower 4bits (4th,3rd,2nd and 1st)carry less than 6%of the image informa-tion,according to Eq.(2).The percentage of the pixel information is shown in Table 1.p ei T?2i

P i ?0

2e2T

In order to reduce the computation cost,we permute the higher 4bits binary images independently and the binary images of the lower 4bits are permuted as a whole to reduce the execution time.

2.2.Self-adaptive encryption

The basic idea of self-adaptive is to divide an image into two parts equally,and the information of one part is used to encrypt the other part.In this paper,we divide an image into two binary parts equally,that is the higher 4bits binary images part (Pic 5–Pic 8)and the lower 4bits binary images part (Pic 1–Pic 4).We use the information of the lower 4bits binary images part to encrypt the higher 4bits binary images part.2.3.Coupled map lattice

A coupled map lattice (CML)is used as a spatiotemporal chaos system here,which is a dynamic system with discrete-time,discrete-space and continuous states.It consists of nonlinear maps located on the lattice sites,named as local maps.Each local map is coupled with other local maps in terms of certain coupling rules.Because of the intrinsic nonlinear dynamics of each

local

Fig.1.Plain Lena image of size 256?

256.

Fig.2.Image encryption experimental results.(a)Pic 1,(b)Pic 2,(c)Pic 3,(d)Pic 4,(e)Pic 5,(f)Pic 6,(g)Pic 7and (h)Pic 8.

Table 1

Percentage of pixel information contributed by different bits.

Bit position (i t1)in the pixel Percentage p (i )of the pixel information 10.392220.78433 1.56864 3.1375 6.275612.55725.108

50.20

L.Teng,X.Wang /Optics Communications 285(2012)4048–40544049

map and the diffusion due to the spatial coupling among the local maps,a CML can exhibit spatiotemporal chaos.CML was intro-duced by Kaneko [11],and can be described as x n t1ei T?e1àe Tf ex n ei TTt

e

2

ef ex n ei à1TTtf ex n ei t1TTTe3T

where n is the time index,i (i ?1,2,y ,L )is the lattice site index,e A (0,1)is a coupling constant;x n (i )represents the state variable for the i th site at time n (n ?1,2,y ).The periodic boundary condition x n (0)?x n (L )is assumed.

The local mapping function f (x )is chosen to be the logistic map:

x n t1?ax n e1àx n T

e4T

with parameter a A (3.5699456,4),and x n A (0,1),the system is in chaotic state.

3.The proposed algorithm

3.1.Higher 4bits binary images permutation algorithm by the CML system

Here we use the information of the lower 4bits binary images part to permute the higher 4bits binary images independently.

Step 1:Without loss of generality,we assume that the size of the plain-image P is M ?N ,convert P into its binary image Pic 1–Pic 8;the size of each image is M ?N .We divide binary images into two parts equally,that is the higher 4bits binary images part (Pic 5–Pic 8)and the lower 4bits binary images part (Pic 1–Pic 4).Converting Pic 1–Pic 8to vectors P i ,each vector has a length of M ?N .

Step 2:Calculate four values’sum (n )(n ?1,2,3,4)by summa-tion of all the binary values within each lower 4bits binary image Pic n .Let g (i ,j )be the binary value at row i ,column j in the binary image,then the value of sum (n )got in image Pic n can be obtained as follows:

sum en T?X

ei ,j TA Pic

n

g ei ,j T:Step 3:The number of CML lattices L is set to 4which is equal to the number of higher 4bits binary images.Initialize the system with the parameters a ,e and value sequences x 0en T?1=eesum en Tt1T?n Ten ?1,2,3,4Tto iterate the chaotic system m tMN times,discard the former m values to avoid harmful effects.Each lattice generates MN values x i (n )(i ?1,2,y ,MN ),so the chaotic system generates 4MN values totally.

Sort each x i (n )value {x m t1(n ),x m t2(n ),y ,x m tMN (n )}in ascending order and get x i en T?x m t1en T,x m t2en T,...,x m tMN en T.Find the position of values x i en Tin x i (n )and mark down the transform positions TN n ?{t 1(n ),t 2(n ),y ,t MN (n )}(n ?1,2,3,4).

Step 4:Rearrange the elements of each higher vector P i (i ?5,6,7,8)according to TN n ,that is,move the t 1(n )element of vector P n t4to the ?rst element,t 2(n )element of vector P n t4to the second element,etc.,until all the elements of each vector have been permutated and get 4vectors P 0i ei ?5,6,7,8T.3.2.Higher 4bits binary images diffusion

Step 1:Use the logistic map (Eq.(4))to generate the chaotic sequence,set the initial parameter a 1and initial value y 0to iterate the chaotic system m 1t4MN times and discard the former m 1values to avoid harmful effects.The chaotic sequence has 4MN elements y (i )?{y 1,y 2,y ,y 4MN }.Select MN elements z (j )(1r j r MN )from y (i ),use the following formula to convert z (j )to integer sequence z 1(j ),

z 1ej T?f loor ez ej T?e1014TTmod 256,

where ?oor (i )is the biggest integer which is not bigger than i .The elements in z 1(j )range from 0to 255.Covert z 1(j )to binary sequence z 01ej T,select the higher 4bits of z 01ej Tand form 4vectors z i (i ?5,6,7,8),each vector having a length of MN .

Step 2:Use the information of the lower 4bits binary images part to diffuse the permuted higher 4vectors P i 0according to the following formula:

C i ?eP i 0tP i à4tz i Tmod 2ei ?5,6,7,8T,

e5T

where C i is ciphered higher 4vector,P i à4are 4lower vectors transformed from 4lower binary images.Covert C i to 4binary images and get ciphered higher 4binary images Pic 0i ei ?5,6,7,8T.

3.3.Lower 4bits binary images permutation

Combine Pic 1–Pic 4horizontally and get matrix P combine with 4N rows and 4N columns.Covert P combine to vector P lower ,which has a length of 4MN .

Sort y (i )?{y 1,y 2,y ,y 4MN }in ascending order and get y 1,y 2,...,y 4MN .Next,?nd the position of values y 1,y 2,...,y 4MN in y (i )and mark down the transform position TM ?{t 1,t 2,y ,t 4MN }.

Rearrange the elements of vector P lower according to TM ,and get permuted vector P 0lower .Covert P 0lower to matrix P 0combine with 4N rows and 4N columns,partition P 0combine into 4matrices Pic 0i ei ?1,2,3,4Tfrom left to right;the size of each matrix is M ?N .Pic 0i ei ?1,2,3,4,5,6,7,8Tare the ciphered binary images,com-bine them to get a integer image (Eq.(1))according to the bit locations within a pixel.So,we can get the encrypted image C pic .

3.4.Image decryption

The decryption procedure is similar to that of the encryption

process but in reversed order.

Set the initial parameters and initial values as same as those in the encryption process and use same methods to obtain TM ,TN i ,and z i .Convert C pic into its binary images Pic 0i ei ?1,2,3,4,5,6,7,8T;the size of each image is M ?N .We divide binary images into two parts equally,that is the higher 4bits binary images part (Pic 05–Pic 08)and the lower 4bits binary images part (Pic 01–Pic 04).Covert Pic 01–Pic 08to vectors C i ;each vector has a length of M ?N .

Combine Pic 01–Pic 04horizontally and get matrix P 0combine ,convert P 0combine to get vector P 0lower .Reverse permute P 0lower according to TM and get the reversed permutation P lower .So,we can get Pic 1–Pic 4.Covert Pic 1–Pic 4to vectors P i (i ?1,2,3,4).

Use the following formula to diffuse C i (i ?5,6,7,8)reversely,where P i 0is the decrypted diffuse value of higher 4binary images 0vectors:

P i 0?eC i àP i à4àz i Tmod2ei ?5,6,7,8T

e6T

Reverse permute P i 0ei ?5,6,7,8Taccording to TN i and get the reversed permutation vectors P i (i ?5,6,7,8).So,we can get Pic 5–Pic 8.Transform Pic 1–Pic 8into the plain image.

4.Experimental results 4.1.Encryption results

The plain-image Lena with size 256?256is used for encryption.The initial parameters and initial values are a ?4.0,e ?0,1a 1?3.99and y 0?0.12345678912345,and m ?m 1?500.Fig.2(a)shows the plain-image.Fig.2(b)shows the encrypted image.

L.Teng,X.Wang /Optics Communications 285(2012)4048–4054

4050

4.2.Security analysis

A good encryption algorithm should resist all kinds of known attacks,such as exhaustive attack,statistical attack and chosen-plaintext/cipher text attack.In this section,we will discuss the security analysis of the proposed encryption scheme.

4.2.1.Key space analysis

A good encryption scheme should be sensitive to the secret keys,and the key space should be large enough to make brute-force attacks infeasible.In this algorithm,the initial conditions and parameters a ,a 1and e ,y 0can be used as keys as well as the initial iteration numbers m and m 1.If the precision is 10à14,the

key space size can reach to 1056;it is bigger than 2128.So,the key space is large enough to resist the brute-force attacks.

4.2.2.Key sensitivity analysis

Several key sensitivity tests are performed.Fig.3(a)shows the encrypted image of Lena with the correct encryption key.And Fig.3(b),(c),(d),and (e)show the encrypted image of Lena with the wrong encryption key a ?3.99999999999999,e ?0.10000000000001a 1?3.99000000000001,and y 0?0.1234567898764respectively.So it can be concluded that the proposed algorithm is sensitive to the key,a small change of the key will generate a completely different decryption result and cannot get the correct

plain-image.

Fig.3.Key sensitivity analysis.(a)Decrypted image with correct initial parameters;(b)decrypted image with different initial value a ?3:99999999999999;(c)decrypted image with different initial value e ?0:10000000000001;(d)decrypted image with different initial value a 1?3:99000000000001and (e)decrypted image with different initial value y 0?0:

1234567898764.

Fig.4.Histograms.(a)Histograms of plain-image and (b)histograms of ciphered-image.

L.Teng,X.Wang /Optics Communications 285(2012)4048–40544051

Fig.5.Correlation of adjacent pixels.(a)horizontally adjacent pixels in the plain image;(b)horizontally adjacent pixels in the ciphered image;(c)vertically adjacent pixels in the plain image;(d)vertically adjacent pixels in the ciphered image;(e)diagonally adjacent pixels in the plain image and (f)diagonally adjacent pixels in the ciphered image.

L.Teng,X.Wang /Optics Communications 285(2012)4048–4054

4052

4.2.3.Histogram analysis

The frequency of each gray level can be seen from the histogram,it leads to the leakage of image information.Simula-tion results are shown in Fig.4,from the?gures,we can see that histogram of the cipher-image is fairly uniform and is signi?-cantly different from that of the original image.

4.2.4.Correlation analysis

In order to protect against statistical attack,we must decrease the correlation of two adjacent pixels in the ciphered image[12]. Calculate the correction coef?cient of each pair by using the following formulas:

r xy?

covex,yT

?????????

DexT

p??????????

DeyT

p,e7T

where

DexT?1

N

X N

i?1

ex iàEexTT2,covex,yT?

1

N

X N

i?1

ex iàEexTTey iàEeyTT,

EexT?1

N

X N

i?1

x i,

here,x and y are gray-scale values of two adjacent pixels in the image.

We randomly select2000pairs of two adjacent pixels from a plain image Fig.1(a)and a ciphered image Fig.1(b)to calculate the correction coef?cient.Fig.5shows the correlation of two images in horizontal,vertical and diagonal directions.Then the correlation coef?cient of the two pixel pairs is calculated in Table2.From Table2we can see that the correlation coef?cient of the ciphered image is much smaller than that of the plain image,so the chaotic encryption algorithm satis?es zero co-correlation.

4.2.

https://www.360docs.net/doc/744635279.html,rmation entropy analysis

Information theory is a mathematical theory of data communica-tion and storage found by Shannon in1949[13].An important theory in information theory is entropy.The entropy H(m)of a message source m can be calculated as

HemT?X

2Nà1

i?0

pem iTlog

1

i

e8T

where p(m i)represents the probability of symbol m i and the entropy is expressed in bits.Actually,given that a real information source seldom transmits random messages,in general,the entropy value of source is smaller than the ideal one.However,when these messages are encrypted,their ideal entropy should be8.If the output of such a cipher emits symbols with entropy of less than8,then,there would be a possibility of predictability which threatens its security.The value obtained is very close to the theoretical value of8.This means that information leakage in the encryption process is negligible and the encryption system is secure against the entropy https://www.360docs.net/doc/744635279.html,ing the above-mentioned formula,we have got the entropy H(m)?7.997450.

4.2.6.NPCR and UACI analysis.

NPCR stands for the number of pixels change rate while one pixel of plain image is changed.The closer NPCR gets to100%,the more sensitive the cryptosystem is to the changing of plain image, and the more effective it is in resisting plaintext attack.UACI stands for the average intensity of differences between the plain image and ciphered image.The bigger the UACI is,the more effective is the cryptosystem in resisting differential attack.Here are the formulae to calculate NPCR and UACI:

NPCR?

P

ij

Dei,jT

W?H

?100%,e9TUACI?

1

W?H

X

i,j

C1ei,jTàC2ei,jT

255

2

4

3

5?100%,e10T

where W and H represent the width and height of the image respectively,and C1and C2are respectively the ciphered images before and after one pixel of the plain image is changed.

Two plain-images are used in the tests.The?rst image is the original plain-image,and the other is obtained by changing the ?rst pixel from‘173’to‘174’.Then the two images are encrypted with the same key to generate the corresponding cipher-images C1and C2.The results according to the proposed algorithm of NPCR and UACI are NPCR?93.6768%and UACI?33.3364%.The results show that the proposed cryptosystem could resist plain-text attack and differential attack effectively.

4.2.7.Classical types of attacks

When cryptanalyzing a cryptosystem,the general assumption made is that the cryptanalyst knows exactly the design and working of the cryptosystem under study;he knows everything about the cryptosystem except the secret key.This is an evident requirement in today0s secure communications networks,usually referred to as Kerckhoff’s principle[14].There are four classical types of attacks:

1.Ciphertext only:the opponent possesses a string of ciphertext.

2.Known plaintext:the opponent possesses a string of plaintext,

and the corresponding ciphertext.

3.Chosen plaintext:the opponent has obtained temporary access

to the encryption machinery.Hence he can choose a plaintext string,and construct the corresponding ciphertext string.

4.Chosen ciphertext:the opponent has obtained temporary

access to the decryption machinery.Hence he can choose a ciphertext string,and construct the corresponding plaintext string.

Obviously,chosen plaintext attack is the most powerful attack. If a cryptosystem can resist this attack,it can resist other types of attack.

In this paper,we use the self-adaptive scheme to encrypt the higher4binary images,that is,use the information of the lower 4bits binary images part to encrypt the higher4bits binary images part.So,different ciphered images have different plain lower4binary images value,the opponent would not be able to use one plaintext and corresponding ciphertext to decrypt the other ciphered image.The proposed algorithm can resist the chosen plaintext/ciphertext attack.

5.Conclusions

In this paper,we proposed a bit-level image encryption algorithm based on spatiotemporal chaotic and self-adaptive Table2

Correlation coef?cient of two adjacent pixels in plain-image and ciphered-image. Correlation coef?cient Plain-image Ciphered-image Horizontal0.9217490.024178 Vertical0.971100à0.019425 Diagonal0.9227490.024322

L.Teng,X.Wang/Optics Communications285(2012)4048–40544053

system.We use a bit-level encryption scheme to reduce the volume of data during encryption and decryption in order to reduce the execution time.We also use the adaptive encryption scheme to make the ciphered image dependent on the plain image,that is,use the information of the lower4bits binary images part to encrypt the higher4bits binary images part,to improve the performance.Simulation results show that the performance and security of the proposed encryption algorithm can encrypt plaintext effectively and resist various typical attacks. Acknowledgment

This research is supported by the National Natural Science Foundation of China(Grant nos,61173183,60973152,and 60573172),the Superior University Doctor Subject Special Scien-ti?c Research Foundation of China(Grant no.20070141014)and the Natural Science Foundation of Liaoning province(Grant no. 20082165).References

[1]S.H.Wang,J.Y.Kuang,J.H.Li,Y.L.Luo,H.P.Lu,G.Hu,Physical Reviews E66(6)

(2002)1.

[2]H.J.Liu,X.Y.Wang,Optics Communications284(2011)3895.

[3]Y.Wang,K.W.Wong,X.F.Liao,G.R.Chen,Applied Soft Computing11(2011)

514.

[4]X.Y.Wang,J.F.Zhao,H.J.Liu,Optics Communications285(2012)562.

[5]A.Kumar,M.K.Ghose,Communications in Nonlinear Science and Numerical

Simulation16(2011)372.

[6]T.Xiang,K.Wong,X.F.Liao,Chaos17(27)(2007)023.

[7]X.F.Liao,https://www.360docs.net/doc/744635279.html,i,Q.Zhou,Signal Processing90(9)(2010)2714.

[8]F.Y.Sun,S.T.Liu,Z.Q.Li,Z.W.L¨u,Chaos,Solitons and Fractals38(3)(2008)

631.

[9]R.Rhouma,S.Meherzi,S.Belghith,Chaos,Solitons and Fractals40(1)(2009)

309.

[10]R.Rhouma,S.Belghith,Chaos17(3)(2007)1.

[11]K.Kaneko,Progress of Theoretical Physics74(5)(1985)1033.

[12]S.Behnia, A.Akhshani,H.Mahmodi,Chaos,Solitons and Fractals35(2)

(2008)408.

[13]C.E.Shannon,Bell System Technical28(4)(1949)656.

[14]N.K.Pareek,V.Patidar,K.K.Sud,Communications in Nonlinear Science and

Numerical Simulation10(7)(2005)715.

L.Teng,X.Wang/Optics Communications285(2012)4048–4054 4054