Encryption Algorithm Based on Logistic Map and Pixel Mapping Table Hazem Al-Najjar Asem Al-Najjar
Encryption Algorithm Based on Logistic Map and Pixel Mapping Table Hazem Mohammad Al-Najjar Technical College of Arar, Department of Computer Technical and Vocational Training Corporation Arar,KSA hazem_najjar@yahoo.com Asem Mohammad AL-Najjar Department of Computer Science Yarmouk University Irbid, Jordan asemalnajjar@yahoo.com Abstract: In this paper, we propose a new image encryption algorithm based on logistic map chaotic function. Our algorithm consists of two replacement approaches; to change the value of the pixel without shuffling the image itself. To do that, we suggest using a Pixel Mapping Table (PMT) with the random shifting value to increase the uncertainty of the image. After that, we modified the pixels value by using the rows and columns replacement approach. However, by analyzing our algorithm, we show that it s strong against different types of attacks and it s sensitive to the initial conditions. Keywords : Encryption, Logistic Map, Pixel Replacement. 1. INTRODUCTION As the internet become very large, the security of digital images and videos become an important issue for all internet users. Because of this, the data between the legitimate users need to be protected before transmission by using encryption methods. The encryption converts the meaningful information into garbage data; so no malicious user can read the data. Moreover, many methods to encrypt the data proposed by researchers such as: RSA, DES and IDEA. On another hand, to secure the images and multimedia application special methods and special rules need to be considered. encryption system used randomization systems; to increase the uncertainty of the encryption process. Chaos is one of the most important theories that used to create a random sequence that firstly used in the computer by Edward Lorenz in 1963. The chaos was used in the encryption system because of its characteristics, like sensitivity to the initial conditions and unpredictability to the chaos sequences. Many approaches try to design image encryption algorithms by using chaos, like [8] used multi-chaotic maps to encrypt the image by dividing his system into mainly two phases, in the first phase pixels are permuted by using a Arnold cat map and in the second phase the muti-chaotic maps are used to encrypt the permuted pixels. In [3] they used two one-dimensional discrete Chebyshev chaotic sequences for row and column scrambling for each pixel on the original image. Where, [1] used Rossler chaotic system to encrypt the image by applying changes in the pixels value and their positions; to increase the uncertainty in the cipher images. The one time pads with the logistic map (as a chaotic function) are used in [5] to encrypt the image and increase the size of the encrypted keys in the cipher. Others, like [2] used a knight s tour with slip encryption filter; to encrypt the image without using any chaotic functions. However, security analysis results, drawbacks and the strength of the chaotic systems are analyzed in [4, 7]. In this paper, we used a logistic map with three different initial conditions to create three different chaotic sequences with two pixels mapping tables to change the pixels value and increasing the uncertainly in the cipher image without shuffling the image itself. In which, this method increase the efficiency of the system and decrease execution time of the algorithm. The rest of this paper is organized as follows. In section II, our approach is described in detail. Experimental results and security analysis are presented in section III. Finally, our conclusions are drawn in section IV. 2. PIXELS-VALUE REPLACEMENT 2.1. Shuffling and Shifting Chaotic Function The logistic map is one dimensional chaotic system; with X output and input variable and two initial conditions X and as follows: X n+1 =λ X n (1-X n ) (1), in which, the chaotic behavior is achieved when as shown in Fig.1. In our encryption algorithm we used a logistic map; to shuffle the Pixels Mapping Tables (PMT) ; to shift the pixels value and to shuffle the pixels value as shown in equation 2. 1.9.8.7.6.5.4.3.2.1 2.5 3 3.5 4 Figure.1: Logistic Map Bifurcation 2.2. Chaotic Pixel Mapping Table Encryption System is divided into two approaches as follows: pixel replacement approach and pixel scrambling approach. In pixel replacement approach we change the pixels value, 56
where, in pixel scrambling we change the pixels position. In this paper, we used only the pixel replacement approach with two Pixel Mapping Table (PMT) that created by using logistic map (Key1). We define a PMT as a table that contains the pixel values from to 255 in the shuffled order with the size 256 x1 as shown in Fig. 2. In which, our algorithm used two PMTs as follows: 1- PMT1: we replaced the pixels value by using PMT1, so the current pixel is not the same after and before the replacement. 2- PMT2: to increase the uncertainty of the cipher image, the column replacement and the row replacement are used. In the first replacement method the two pixels in the original image will be mapped into the same value. So, to solve this problem we used a logistic sequence to randomize the pixels value as shown in the equation 2: Where, i, j is the position of the pixel, random is the random value generated by using a logistic map (Key2). Mod is the modulus operation. The pixel_level represents the number of available colors (our case this value equal 256) and c is the constant value. Therefore, if two pixels have the same value in the original image, the cipher image for two pixels is differing completely. So, the crypto-analysis of the cipher images will not get any useful information about the PMT design. 1 2 3 255 125 2 5 1 Figure.2: Unstuffed and Shuffled PMT, Respectively 2.3. Rows and Columns Replacement After the previous step, another PMT is created (key3) to change the pixel values by using the XOR operation for each column in the cipher image, as follow: (5) Where, k is the column number in the cipher image and PMT2 is the pixel mapping table generated by using a logistic map (key2). Each pixel in the image was replaced by using equation 2 and changed their values by using equation 5. After that, the Rows replacement was used to change the pixels in the image and modified their values by rows as follows: (6) Example: Assume 4x 4 image, 4 bits for the gray level which generates 16 different colors and suppose that there is two pixels mapping tables PMT1 and PMT 2; to replace the pixels in the image as shown in Figs. [3-5]. 15 3 1 6 14 13 8 4 2 12 5 9 7 1 11 8 11 13 15 Figure.3: Pixels Mapping Tables, PMT1 and PMT2, Respectively 1 2 3 1 2 4 3 4 5 6 7 7 14 1 14 8 9 1 11 6 2 55 11 12 13 14 15 9 1 25 15 1- After Applying equation 2, the pixels value will be as shown in Fig. 5 a. 2- Then we map each value in the image by using equation 3 (Fig.5 b). Figure.4: The Original and the Shifting, Respectively. 1 5 1 1 1 1 5 4 1 1 6 6 1 7 14 ( b) Figure.5: after Using Equation 2 and 3, Respectively. 3- Rows and columns replacement: the pixels will be changed by using the Xoring operation with PMT2 (shown in figure.3 a), in which, the PMT in fig. 3 b will be used to change the values of columns and rows, respectively. 2.4. Encryption Scheme Diagram Our algorithm used only replacement approaches to encrypt the image. The two replacements approaches are used: in the first approach, we shift the pixels by using a random value and mapping it by using PMT. Where, in the second method, we used replacement by using the Xoring operation with specific random vector generated by using a logistic map (as shown in the fig 6). However, the decryption process is done in the reverse order. Key 1 Shifting + Modulus Operations Column s Replacement Key 3 12 14 12 12 15 14 6 13 13 8 1 Key 2 PMT Row s Replacement Shuffled Figure.6: Encryption Algorithm Diagram 57
3. EXPERIMENTAL RESULTS AND SECURITY ANALYSIS In our experiment, we try to evaluate our algorithm in encrypting the images and the ability of the attackers to break it by using the different analysis such as histogram analysis, key space analysis entropy analysis and correlation analysis and other types of analysis. Moreover, Lena and Cameraman images are used as tested images with the size 256x256. The cipher and original images of the Lena and Cameraman are shown in Fig.7, respectively. With input keys Key1= 1x1 Key2= 2x1-15 and Key3= 2x1-13 for two images. Lena Cameraman Figure.8: Sensitivity Tests of Keys 3.3. Information Entropy Analysis Entropy determines the randomness of system, where, the true random variable should generate 2 8 symbols with equal probability and the entropy value equal 8. To evaluate the entropy and calculate the entropy value, we used a following equation [6]: Where P (S i ) represents the probability of symbol S i, in our tests the average entropy of the lena cipher image is 7.9996 and for the cameraman cipher image is equal to 7.9997, which are very close to the optimal value, then the entropy attack is not possible. (c) (d) Figure.7: Lena and Cameraman and Their Ciphers s, Respectively 3.1. Keys Space Analysis The key space of our algorithm is depending on the three keys of the logistic map function. However, the key space is calculated as follow: we have three keys key1, key2 and key3, the key space of each one is equal to 1 15 then the key space of the algorithm is equal to 1 45. 3.4. Histogram Analysis By analyzing the image histogram the cryptanalyst can get very useful information from the cipher image. In which, the good encryption algorithm should generate uniformly distribution of the histogram. In our tests, it s very difficult to get any information from the histogram; Fig. 9 shows the histogram analysis of lena and cameraman image and there cipher images, respectively. 1 8 6 4 6 5 4 3 2 3.2. Keys Sensitivity Analysis The encryption system should be sensitive to the small changes on the decrypted keys. And, generate a wrong decrypted image, if there is a small difference in the decryption keys. Our sensitive tests keys Key1= 2x1-14, Key2= 8x1-15 and Key1= 5x1-13, in which, Fig.8 shows the decrypted image for the lena and cameraman tested images by using a wrong decryption keys. 2 5 1 15 2 25 1 5 1 15 2 25 58
8 7 6 5 4 3 2 1 6 5 4 3 2 1 (Unified Average Changing Intensity). Where, NPCR defined as a percentage of different pixels number between two cipher images and UACI defined as an average intensity of differences between two cipher images as defined in the following[6]: 5 1 15 2 25 5 1 15 2 25 (c) (d) Figure.9: Histogram of Lena and Cameraman and Cipher s, Respectively. 3.5. Correlation Analysis In the correlation analysis the adjacent pixels will be tested to see the correlation between them, if the algorithm has a very large correlation then the algorithm may be broken by using a correlation analysis. For this, we try to test our system by calculating the correlation coefficient for all possible cases in vertical, horizontal and diagonal adjacent. Where, the correlation coefficient is calculated by using the following formula [6]: Where M x N is the size of the cipher images and C1 and C2 are two different cipher images encrypted by using a different keys, where D(i, j) is defined as follow: After calculations, we get the Average NPCR and UACI of lena image are: NPCR = 99.6231and UACI = 33.47 and that of the cameraman are: NPCR = 99.6429 and UACI = 33.5644. Then our algorithm has a good ability against known plain text attack. Where, M is the total number of randomized pairs, i and j are the two vectors that contains i values and j values of the pair in the tested image, respectively. Table.1: Correlation Coefficients of Adjacent Pixels Lena Cameraman Coefficient Plain Cipher Plain Cipher Vertical.969.186.967 -.396 Horizontal.9278.965.9381 -.592 Diagonal.96.161.9115.56 Table.1 Shows the correlation coefficients between two adjacent pixels in all possible cases (vertically, horizontally and diagonally) of the plain-text images and cipher images. The results revealed that the proposed method randomized the pixels in good way. 4. CONCLUSION In this Paper, a new approach to encrypt the image by using a Logistic map with the pixel mapping tables is proposed. In which, the algorithm consists of two replacement methods without any scrambling approach which enhance the execution time of the encryption algorithm. In the first method, we shift each pixel by using a random shifter generated by using the logistic map as a Key1 and modified by using the modulus operation. The resulted image will be mapped by using PMT (key2) and modified by using another PMT (key3) for each column and row in the image; to enhance and increase the uncertainty of the cipher image. However, we shown by experimental results that our algorithm is sensitive to initial conditions and strong against the brute force attacks. Finally, we found that our algorithm has a high security against different types of attacks with the large space of the encryption keys. 3.6. Plain Text Sensitivity Analysis If the cipher image is not sensitive in the changing of the plaintext then the cryptanalyst can get very useful information from the encrypted images. For this, we use two criteria, NPCR (Number of Pixel change Rate) and UACI 59
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