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An improved Image Encryption Scheme Based on Line Maps

Juan Li, Yong Feng, Xuqiang Yang

Department of Electrical Engineering

Harbin Institute of Technology

Harbin, china

lijuan2001422@163.com

Abstract—In order to avoid the flaw while keeping all the

merits of image encryption scheme based on line maps, an

improved image encryption scheme is proposed, which

extended the original 2D scheme to 3D. An image with size

N×M is firstly depicted by a 3D bit matrix, and then it is

supposed as composed of 8×M vectors with size N. Line

maps are used to stretch the vectors to an array, while the

fold map is used to transform the array to a same sized 3D

matrix. Simulation results show that the improved image

encryption scheme complete pixel permutation and

confusion simultaneously, it enhance the security of the

original cipher.

Keywords- Cryptography ; image encryption; Line maps

I.INTRODUCTION

With the great advances in digital image, information

storage and Internet technologies, more and more digital

images are transmitted over the Internet. Some digital images

may relate to the personal privacy, company benefit, military

secret or state security, therefore security of digital images

has increasingly become an important issue.

Compared with plain image, the ciphered image is

generally changed in three aspects: pixel positions, pixel

values and correlation coefficient between two adjacent

pixels. Recently, the researchers mainly concentrate on the

following technologies:

(1) Image encryption technologies in space domain

The method includes permutation based image

encryption scheme and information entropy based image

encryption scheme. The image encryption schemes based on

game of life, knight-tour transformation[1], Cat map[2] are

belong to permutation based image encryption scheme. Cat

map is a two-dimensional chaotic map introduced by Arnold

and Avez. Guanrong Chen[3,4] Zaiguang Ma .etc developed

cat map, proposed several 2D and 3D image encryption

schemes.

Information entropy based image encryption scheme

could degenerate the correlation between two adjacent pixels,

consequently change the information entropy and histogram

of plain image. The substitution changes each pixel value in

the plain-image one by one which generally adopts XOR

operation[5], while the diffusion change several adjacent

pixels at one time.

(2) Image encryption scheme in transform domain

Digital image has inner features such as large data

amount, high redundancy and strong correlative. In favor of

being stored or transmitted, the digital image is usually

preprocessed by a kind of orthogonal transformation first,

and then encoded. If encryption algorithms are implemented

with the encoding operation at the same time, this scheme is

called image encryption scheme in transform domain. Tree

structure and SCAN language based image encryption are

both image encryption schemes in transform domain. Binary

tree and quad tree structure are broadly used in image

encryption[6,7]. SCAN based image encryption scheme is

proposed by N.Bourbakis and S.S. Maniccam[8].

Based on the image encryption scheme using Line maps[9]

proposed by Professor Feng, An improved 3D image

encryption is proposed in this paper. A 256 grey-scale image

of size M×N is described by a 3D bit matrix with M×N×8

dimensional, then this 3D matrix is permutated according to

a sequence of iterations of Line maps and fold maps

determined by the key. At last it is transformed back to a

same sized 3D matrix, thus ciphered image could be

obtained. Image decryption is the inverse process of image

encryption.

II.IMAGEENCRYPTION APPROACH BASED ON LINE MAPS

The Image Encryption Approach based on Line maps is

new invertible two-dimensional map, it maps a square

image to an array of pixels and then, maps it back from the

array to a same sized square image. A Line map consists of

two submaps: the left Line map and the right Line map. For

image encryption and decryption, the security key is used to

represent a sequence of numbers of iterations of the left and

the right Line maps alternatively.

The working principle of Line map is shown in Figure.1.

The left Line map projects an N?N square image to an array

of N?N pixels from the upper left corner to the lower right

corner along the diagonal. Then the array of N?N pixels is

further mapped to a same sized square image. As shown in

Figure.1(b), the right Line map is symmetric to the left Line

map.

The Image Encryption Approach has several advantages

as shown in [9]. However, it only changes the pixel position,

if used to encrypt image without substitution, it is not secure

against chosen-plaintext attacks. To improve the security of

2009 Fifth International Conference on Information Assurance and Security

978-0-7695-3744-3/09 $25.00 © 2009 IEEE

DOI 10.1109/IAS.2009.67

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the original cipher, an improved 3D image encryption

scheme is proposed in this paper.

(a)

(b)

Figure.1 The process of Line maps : (a) The left Line map

(b) The right Line map

III.IMPROVED 3D ENCRYPTION SCHEME

For a grey-scale image, it is generally described by a 2D

digital image. In this paper, a 256 grey-scale image of size

M×N is taken as an example to illustrate the image

encryption scheme. 8 binary bits are utilized to denote each

decimal pixel values in the 2D matrix, therefore the grey-

scale image of size M×N is transformed to a 3D matrix with

M×N×8 dimensional.

Take the image of size 3×3 as an example to describe the

process of 3D invertible map. The image of size 3×3 could

be mapped to a 3D bit matrix with 3×3×8 dimensional. This

3D digital matrix is then assumed as consist of 24 vectors,

denoted by

( )l

, where

1,2,3i ?

, each vector has 3 elements, as depicted in

Figure.2.

kji

1,2,,8k ?

?

?

1,2,3j ?

?

Figure.2 The 3D matrix is consist of 24 vectors

The 24 vectors are mapped to an array according to the

algorithm shown in Figure.3. Each vector from super

diagonal is inserted between the adjacent two vectors of the

lower level adjacent super skew diagonal. For example

vector 11 l is inserted between vector

3 vectors are linked to a new vector from beginning to end.

Vector

31

l is inserted between vector

they are linked from beginning to end. Similar operations

21

l and 12 l . Then these

41

l and

32

l , and then

are carried out for the left vectors, 5 new vectors could be

obtained, at last, they are also linked to an array of 72

elements from beginning to end subsequently, denoted by l .

This process is called left Line map in this paper.

The other method of mapping the 24 vectors to an array

is called right Line map which is symmetric to left line map,

as shown in Figure.4.

11l

21 l

31 l

41 l

51 l

61 l

71 l

81l

12l

22 l

32 l

42 l

52 l

62 l

72 l

82l

13l

23 l

33 l

43 l

53 l

63 l

73 l

83l

Figure.3 Left Line map

11l

21 l

31 l

41

l

51 l

61 l

71 l

81 l

12l

22 l

32 l

42 l

52 l

62 l

72 l

82 l

13l

23

l

33 l

43

l

53 l

63 l

73 l

83 l

Figure.4 Right Line map

The array obtained above is mapped to a same sized 3D

matrix as shown in Figure.5. For example, the array of size

72 is firstly divided to 9 same sized vectors with length 8,

then the 9 vectors constitute a 3D matrix, this process is

called fold map. Therefore, through either the left Line map

or the right line map and fold map, one iteration of 3D

invertible map has been completed.

Figure.5 The process of Fold map

The algorithms of image encryption and decryption are

formulated. Because of space-saving, they won’t be given

here.

IV.

In order to test the improved 3D invertible image

encryption proposed, image encryption and decryption

simulation are carried out. For an image of size 256?256,

image encryption and decryption are carried out with the

security key=1234567890123456, the results are shown in

Figure.6. Since the decrypted image is same with plain

image, it won’t be shown here.

SIMULATION RESULTS AND SECURITY ANALYSIS

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(a) (b)

Figure.6 Plain image and ciphered image: a) Plain image;

b) Ciphered image

Aiming at testing the security of this improved 3D image

encryption scheme, the histogram, key sensitivity, key space,

unchanged point, information

coefficient are analyzed.

Key sensitivity analysis is firstly carried out. For the

ciphered image encrypted with Key1=123456789-0123456,

shown in Figure.6 a), now it is decrypted by Key2=

1234567890123457 and Key3=223456789012-

respectively, the decrypted images are depicted in Figure.7.

From Figure.7, it can be seen that neither Key2 nor Key3

could correctly decrypt the ciphered image, although there is

only one digit difference between Key1 and Key2, Key3

and Key1.

Each digit of security key is an integer between 0 and 9,

the key space is only relate with the length of key, the key

space is shown in Table.1. In theory, the key could be an

integer of any length which is suitable to a wide range of

security requirements, therefore the key space of this 3D

image encryption scheme could be very large, ensure the

security of the ciphered image.

entropy, correlation

3456

(a) (b)

Figure.7 Decrypt the ciphered image: (a) Decrypt with Key2;

(b) Decrypt with Key3

TABLE I.

KEY SPACE OF THE 3D INVERTIBLE MAP

Key length (bits)

16 32 64 128 256

Key space

16

10

32

10

64

10

128

10

256

10

Since the 3D invertible image encryption scheme

permute the binary bits of the pixels in the image, it equals

to change the pixel positions and values in plain image,

namely it realize permutation and confusion simultaneously,

as a result, the histograms of plain image and ciphered

image are different, shown in Figure.8.

From Figure.8, it can be seen that the histogram of

ciphered image is fairly uniform, this due to that the 3D

image encryption scheme permute the binary bits of pixels

in plain image which greatly degenerate the correlation

coefficient between two adjacent pixels.

050 100150 200 250

0

100

200

300

400

500

600

700

800

0 50 100150200 250

0

100

200

300

400

500

600

(a) (b)

Figure.8 Histograms of two images: (a) Histogram of plain image;

(b) Histogram of ciphered image

To test the correlation coefficient between two adjacent

pixels, 4000 pairs of two horizontally adjacent pixels, two

vertically adjacent pixels and two diagonally adjacent pixels

are randomly selected from the plain image and ciphered

image respectively. Then, calculate their correlation

coefficients. Table.2 shows the correlation distribution of

two horizontally adjacent pixels in the plain-image and that

in the ciphered image. From Table.2, it can be seen that the

correlation coefficient of two adjacent pixels in plain image

is close to 1, while in ciphered image it is close to 0. It

demonstrates that the image encryption scheme could

effectively resist statistical attack.

Suppose the pixel position in plain image A is (i?j), if its

value is not changed after image encryption operation, it is

called unchanged point. The percentage of the number of

unchanged points to all of the pixels in an image is called

the percentage of unchanged point, denoted by BD(A).

According to its formula, the percentage of unchanged point

of plain image, shown in Figure.6 a), is 0.358%. There are

few unchanged points in plain image and the image

encryption algorithm changes above 99.6% pixel positions

in plain image. It means that this image encryption

algorithm effectively permute the plain image.

TABLE II. CORRELATION COEFFICIENTS IN TWO IMAGES

Direction Plain image Ciphered image

horizontal 0.9428 0.0013

vertical 0.9713 0.0061

diagonal 0.9412 0.0026

Information entropy indicates the distributing of the pixel

grey-scale values in the image. The larger the information

entropy is, the more the uncertain information in the image

there is. According to the formula, the information entropy

of plain image, shown in Figure.6 (a), is 7.4255, while the

information entropy of ciphered image, shown in Figure.6

(a), is 7.9889, which is much larger than that of plain image

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and also close to the maximum of information entropy

which is 8. It indicates that the pixel grey-scale values

distributing in ciphered image is quite uniform, little

information could be obtained from the ciphered image

which enhances the attack difficulty.

V.

CONCLUSION

An improved 3D invertible image encryption algorithm

based on Line maps is proposed in this paper. The image

encryption in [9] only changes the pixels positions, so when

it is used to encrypt image, substitution should be adopted.

While the improved encryption scheme realizes permutation

and confusion via one iteration of Line map and fold map,

which could enhance the security of the original cipher;

meanwhile, it keeps all the merits of Line maps, such as, has

large key space, high key sensitivity.

ACKNOWLEDGMENT

This work was supported by the National Natural Science

Foundation of China (No. 60474016 and No. 60774040).

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