2017 2nd International Conference on Advances in Management Engineering and Information Technology (AMEIT 2017) ISBN: 978-1-60595-457-8
A Novel Image Encryption Algorithm Based on Bernoulli Maps
Chun-hu LI
*, Guang-chun LUO, Ke QIN and Chun-bao LI
School of Computer Science and Engineering
University of Electronic Science and Technology of China
Chengdu, Sichuan, 610054, P.R.C
*
Corresponding author
Keywords: Image encryption, Bernoulli maps, Chaos.
Abstract. This paper presents a novel image encryption algorithm, which is based on the Bernoulli
maps. Image encryption systems based on Bernoulli maps shows some better performances. We also use the well-known ways to perform the security and performance analysis of the proposed image encryption algorithm. The result of the experiment analysis proves that the proposed scheme is efficient and secure.
Introduction
In recent years, a growing number of discrete chaotic image encryption algorithms have been proposed [1-7]. However, most of them encounter some problems such as the lack of robustness and security. In this paper, we introduce a novel image encryption algorithm based on coupled Bernoulli maps. This system is a chaotic image encryption system. Owing to the exceptionally desirable properties of mixing and sensitivity to initial conditions and parameters of chaotic maps, chaos-based encryption has suggested a new and efficient way to deal with the intractable problem of fast and highly secure image encryption. This algorithm tries to improve the problem of failure of encryption such as small key space, encryption speed and level of security.
Bernoulli Maps
Each elemental map is the Bernoulli maps given by Eq. 1.[8]
( )
( )
2
1 1, 1
1
1 2
1 1, 1
1
{
n nn n
X X
n
X X
X
+ − − < < ∆ + ∆
+
− + ∆ < < − ∆
=
(1)
Where, ∆ is a parameter satisfying -1<∆<1. (Fig. 1) This map is so simple that many quantities can explicitly be calculated. These quantities imply highly chaotic behavior of map Eq. 1. The Bernoulli map Eq. 1 is therefore equivalent to the coin toss with events represented by 1 and -1.
Proposed Cryptosystem
In this section, we use the Bernoulli maps Eq. 1 to implement encryption process. The image encryption algorithm proposed in this paper consists of the following major steps:
Step 1) Read plain-images (original-image) (M a×b ), get size of M, e.g. use [a, b, c] to save the size
of M, a=256, b=256, c=3, let N=a*b*c, and initiate control parameter (∆) of the Bernoulli maps; Step 2) Input the secret (encryption) key X0 into the algorithm. Iterate the Bernoulli maps N times
using Eq. 1, and obtain the key array X(N), the size of X(N) is N;
Step 4) The resulting image of step (3) is the ciphered image. The image encryption scheme is shown in Fig. 3.
The image decryption algorithm is the inverse process of encryption.
[image:2.612.88.539.190.619.2]The following experimental results are presented to demonstrate the efficiency of the proposed image encryption algorithm. The standard gray scale image Lenna (Fig. 2(a)) with the size 256 × 256 pixels is used for this experiment. The results of the encryption are presented in Fig. 2(b). As can be seen from the figures, there are no patterns or shadows visible in the corresponding cipher image. The results of the decryption are presented in Fig. 2(c). As can be seen from the figures, there is no different from the original image.
Figure 1. Bernoulli map.
Figure 2. (a) The original image. (b) The encrypted image. (c) The decrypted image. (d) The histogram of original image. (e) The histogram of ciphered image.
Performance and Security Analysis
Key Space Analysis
T (X0 , ∆) = θ(X0 ×∆), (2)
Where X0∈ (-1, 1), ∆∈ (-1, 1), the each precision of X0 and ∆ is 10−16 , namely, the size of key
space is ( 1016)2 ). This key space is big enough for brute-force attacks [9].
Correlation Analysis of Two Adjacent Pixels
The superior confusion and diffusion properties are shown by a test on the correlation of adjacent pixels in the ciphered image. We have also analyzed the correlation between adjacent pixels in ”Lenna” ciphered image. We have used Eq. 2 to calculate the correlation coefficient in the horizontal, vertical and diagonal, the following relation has been used [10]:
(
)
( )
(
)
( )
(
)
1 1 1
2 2
2 2
1 1 1 1
N N N
j j j j
j j j
r
N N N N
j j j j
j j j j
N x y x y
C
N x x N y y
= = = = = = = − = − −
∑
∑
∑
∑
∑
∑
∑
(3) where xj and yj are the values of the adjacent pixels in the image and N is the total number of pixels [image:3.612.85.526.352.404.2]selected from the image for the calculation. We have chosen randomly 1000 image pixels in the original image and the ciphered image respectively to calculate the correlation coefficients of the adjacent pixels in diagonal, horizontal and vertical direction (In Table 1). It demonstrates that the encryption algorithm has covered up all the characters of the original image showing a good performance of balanced 0 1 ratio.
Table 1. Correlation coefficient of two adjacent pixels in simulated original and ciphered images.
Direction Original image Ciphered image
Horizontal 0.9453 0.0021
Vertical 0.9278 0.0027
Diagonal 0.8965 0.0018
Information Entropy Analysis
Information theory is a mathematical theory founded in 1949 by Shannon .Modern information theory is concerned with error-correction, data compression, cryptography, communications systems, and related topics. There is a well-known formula for calculating this entropy[11]:
( )
(
)
(
)
2 1 2 0 1 lo g N i i iH s P s
P s −
= =
∑
(4) where P(Si) represents the probability of symbol Si and the entropy is expressed in bits. The ideal
entropy value for an encrypted image should be 8. The calculation of entropy for the ciphered is presented below:
( )
( )
( )
255 2 0 1 log 7.9998956 i i iH s P s
P s
=
=
∑
=(5) The result shows that the entropy of the encrypted image is very close to the ideal entropy value, which is 8 higher than most of the other existing algorithms. This indicates that the rate of information leakage from the proposed image encryption algorithm is close to zero.
Speed Performance
Table 2. Average ciphering time taking of different size images.
Images size (in pixels) Bits/pixels Average ciphered time(s)
256 × 256 24 1.15-1.23
512 × 512 24 1.98-2.16
1024 × 1024 24 15.96-18.05
2048 × 2048 24 65.37-67.89
Distribution of the Ciphertext
An image histogram displays that how pixels in an image are distributed by plotting the number of pixels. Here taking a Lenna image as the original image, its size is (256 × 256). Histogram of the original image and the corresponding ciphered image are shown in Fig. 2(d) and Fig. 2(e). As it was displayed, the histograms of the original image and the ciphered image, it does not provide any clues to the use of any statistical analysis attack on the encrypted image.
Summary
Nowadays, it is very important to provide a secret image. In this paper, we proposed a new image encryption scheme. The experiment has been presented and security analyzed. In short, the 3D piecewise chaotic maps, which are good for image encryption, may be suitable for generating an image of secret data. The image encryption technique is developing very fast, it needs to continue to propose a new encryption algorithm.
Acknowledgement
This research was financially supported by the foundation of science and technology department of Sichuan province No. 2015SZ0231
References
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