An improved Image Encryption Scheme Based on Line Maps
Juan Li, Yong Feng, Xuqiang Yang
Department of Electrical Engineering
Harbin Institute of Technology
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
Keywords- Cryptography ; image encryption; Line maps
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
(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, Cat map 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
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, 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.
Based on the image encryption scheme using Line maps
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
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
The Image Encryption Approach has several advantages
as shown in . 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
the original cipher, an improved 3D image encryption
scheme is proposed in this paper.
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
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,
, each vector has 3 elements, as depicted in
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.
l is inserted between vector
they are linked from beginning to end. Similar operations
l and 12 l . Then these
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.
Figure.3 Left Line map
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
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
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
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.
Figure.7 Decrypt the ciphered image: (a) Decrypt with Key2;
(b) Decrypt with Key3
KEY SPACE OF THE 3D INVERTIBLE MAP
Key length (bits)
16 32 64 128 256
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 50 100150200 250
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
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.
An improved 3D invertible image encryption algorithm
based on Line maps is proposed in this paper. The image
encryption in  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.
This work was supported by the National Natural Science
Foundation of China (No. 60474016 and No. 60774040).
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