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Extended substitution–diffusion based image cipher using chaotic standard map
Commun Nonlinear Sci Numer Simulat 16 (2011) 372–382
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Extended substitution–diffusion based image cipher using chaotic standard map Anil Kumar *, M.K. Ghose Computer Science and Engineering Department, Sikkim Manipal Institute of Technology, Sikkim, India
a r t i c l e
i n f o
Article history: Received 25 September 2009 Received in revised form 28 January 2010 Accepted 12 April 2010 Available online 20 April 2010 Keywords: Chaotic standard map Diffusion Encryption
a b s t r a c t This paper proposes an extended substitution–diffusion based image cipher using chaotic standard map [1] and linear feedback shift register to overcome the weakness of previous technique by adding nonlinearity. The first stage consists of row and column rotation and permutation which is controlled by the pseudo-random sequences which is generated by standard chaotic map and linear feedback shift register, second stage further diffusion and confusion is obtained in the horizontal and vertical pixels by mixing the properties of the horizontally and vertically adjacent pixels, respectively, with the help of chaotic standard map. The number of rounds in both stage are controlled by combination of pseudo-random sequence and original image. The performance is evaluated from various types of analysis such as entropy analysis, difference analysis, statistical analysis, key sensitivity analysis, key space analysis and speed analysis. The experimental results illustrate that performance of this is highly secured and fast. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Security of multimedia data is most important days due to the widespread transmission over various communication networks. It has been observed that the traditional encryption schemes [2] fail to protect multimedia data due to some special properties and some specific requirements of multimedia processing systems, such as mammoth size and strong redundancy of uncompressed data. The fundamental features of chaotic systems such as ergodicity, mixing property, sensitivity to initial conditions, system parameters and which can be considered analogous to ideal cryptographic properties such as confusion, diffusion, balance and avalanche properties. Hence, many chaos-based encryption systems also proposed [1,3–6]. Many chaos-based cryptography schemes have been successfully cryptanalyzed [7–9]. In this paper, we have proposed an extended symmetric image encryption by using the chaotic 2D standard map and linear feedback shift register which in turn introduces the nonlinearity which is the main limitation of the Patidar et al. [1]. The proposed algorithm comprises three stages, viz. (i) Generation of pseudo-random sequences and synthetic images. (ii) Horizontal, vertical rotation and permutation and XORing with synthetic image. (iii) Vertical, horizontal diffusion and XORing with synthetic image. Pseudo-random sequences is generated by linear feedback shift register and standard chaotic map, while the synthetic images are generated using standard chaotic map. Total number of rounds for two stages are calculated using pseudo-random sequence which is generated using standard chaotic map and original image. * Corresponding author. E-mail address: [email protected] (A. Kumar). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.04.010
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The rest of the paper is organized as follows: In Section 2, brief description chaos, linear feedback shift register and of the Patidar [1] cryptosystem. The detailed algorithms of the proposed encryption procedures is discussed in Section 3. In Section 4, analysis of the proposed image cipher and evaluate its performance through various statistical analysis, key sensitivity analysis, differential analysis, key space analysis, speed analysis, etc. Finally, Section 5 concludes the paper.
2. Basic of chaos, LFSR and brief discussion of cryptosystem [1] 2.1. Chaotic sequence It has a long circle, sensitive to initial value, and unpredictability. The encryption with chaos is fast, so it is widely combined with traditional encryption. The general chaotic system model is given as below
xðnÞ ¼ f ðxðn 1ÞÞ
ð1Þ
where the x(n) is a chaotic sequence generated by the nonlinear f(.), x(0) are the initial condition values. The 1D logistic maps whose chaotic intervals are both in [0, 1] as follows
X nþ1 ¼ X n lð1 X n Þ
ð2Þ
where l[0, 4] and X[0, 1] The compound chaotic functions whose chaotic intervals are both in [1, 1] as follows
( FðxÞ ¼
8x4 8x2 þ 1 . . . ; 4x3 3x . . . ;
x<0
x P 0:
ð3Þ
The chaotic standard map have a large key space compared to another maps Lian et al. [10].
X nþ1 ¼ X n þ K sinY n
ð4Þ
Y nþ1 ¼ Y n þ X nþ1
where Xn and Yn are taken modulo 2p and K is constant. We will generate the pseudo-random sequences using Eq. (4) which in turn used for generating the synthetic images and in calculating the numbers of rounds.
2.2. Linear feedback shift register A Linear feedback shift register (LFSR) is a mechanism for generating binary sequences [11]. Fig. 1 shows a general model of an n-bit LFSR. LFSR are extremely good pseudo-random binary sequence generators [11]. When this register is loaded with any given initial value (except 0 which will generate a pseudo-random binary sequence of all 0s) it generates pseudo-random binary sequence which has very good randomness and statistical properties. The only signal necessary for the generation of the binary sequence is a clock pulse. With each clock pulse a bit of the binary sequence is produced. A example of 4-bit LFSR is considered to demonstrate the functioning of LFSR with the feedback function f = 1 + x + x4. Its initial bit values are used (1111). The output sequence Zn: 011111000000001. . . Generated by LFSR in is periodic of period 15. Period of the sequence generated by LFSR is maximum if we use the primitive polynomial. To design any stream cipher system, one needs to consider the LFSR with primitive feedback polynomials as the basic building blocks. The pseudo-random sequence depends upon the initial seed value and feedback function.
Fig. 1. A general model of n bit linear feedback shift register.
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2.3. Brief description of cryptosystem [1] Patidar et al. [1] proposed a cryptosystem which was utilizes the chaotic 2D standard map and 1D logistic map and which is specifically designed for the coloured images. The initial condition value, number of iterations together consider as the secret key for the algorithm. The algorithm comprises four rounds: two for the substitution and two for the diffusion. The first round of substitution/confusion is achieved with the help of intermediate XORing keys calculated from the secret key. Then two rounds of diffusion namely the horizontal and vertical diffusions are completed by mixing the properties of horizontally and vertically adjacent pixels, respectively. In the fourth round, a robust substitution/confusion is accomplished by generating an intermediate chaotic key stream (CKS) image in a novel manner with the help of chaotic standard and logistic maps. Rhouma et al. [8] shows that all the steps are linear in nature and Rhouma also shows that this cryptosystem vulnerable to attacks by dividing the cryptosystem procedure in two majors successive steps: (1) the diffusion process and (2) the masking process. 3. Proposed method In the encryption, it includes the input image, secret keys. First, generation of pseudo-random sequences and synthetic images using standard chaotic map explained in Section 3.1, calculation of the numbers of rounds in Section 3.2, generation of pseudo-random sequence using LFSR explained in Section 3.3, rotation and permutation and XORing with synthetic image explained in Section 3.4, diffusion and XORing with synthetic image explained in Section 3.5, and decryption discussed in Section 3.6. 3.1. Generation of synthetic images and pseudo-random sequence using standard chaotic map Here, generation of synthetic images and pseudo-random sequences is discussed. Refer Algorithm 1. As we know that conversion floating-point number to integer is required for the chaos based encryption, this conversion and multiplication are time consuming compare to modulus, addition, XOR, etc. operations [13]. Hence, in the proposed method for generation of the synthetic images floating-points to integer conversions and multiplication is reduced into order decrease the computational complexity. 1. The secret key consists of three floating-point numbers and one integer (x0, y0, K, N), where x0, y0 2 (0, 2p), K can have any real value greater than 18.0 and N is any integer value, ideally should be greater than 100. 2. By iterating Eq. (4) W * H + N times, the pseudo-random sequence generated as XKey and YKey by taking last 256 values. 3. The synthetic images and scaled pseudo-random sequence are explained in Algorithm 1.
Algorithm 1. Synthetic images generation and scaled pseudorandom sequence for i = 1 to 256 j k X1ðiÞ ¼ XKeyðiÞ 256 ; 2p j k Y1ðiÞ ¼ YKeyðiÞ 256 ; 2p end for i = 257 to W * H X1ðiÞ ¼ Y1ðX1ði 256ÞÞ bsðY1ðX1ði 128ÞÞÞ bsðY1ðX1ði 64ÞÞÞ; //Where bs is the byte substitution taken from AES Y1ðiÞ ¼ XKey1ðY1ði 256ÞÞ bsðX1ðY1ði 128ÞÞÞ bsðX1ðY1ði 64ÞÞÞ; end SImage1 = reshape (X1,W,H); SImage2 = reshape (Y1,W,H); 3.2. Numbers of rounds The numbers of rounds for both rotation and diffusion is depends upon the pseudo-random sequence generated using standard chaotic map and original image. Refer Algorithm 2. Algorithm 2. Numbers of rounds 1. 2. 3. 4.
Read the image, let the size of image is W * H. XOR all the bytes of the image let it be IXOR. XOR of the values of the pseudo-random sequence X1 as KXORX and of Y1 as KXORY. The number of the rounds for rotation NR = mod(XOR(IXOR, KXORX), 5) + 1. The number of the rounds for diffusion ND = mod(XOR(IXOR, KXORY), 5) + 1.
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3.3. Generation of pseudo-random sequence using LFSR Here, the pseudo-random sequence generated using LFSR for all values (1, 256) considering the size of image is 256 256. Considering the original image size is 256 256. Generate the pseudo-random sequences using the primitive root x8 + x6 + x5 + x4 + 1. Refer Algorithms 3 and 4. Algorithm 3. Function to calculate pseudo-random sequence generation using LFSR function D = LFSR (input) for i = 1 to 256 if i = 256 D(i) = 256; else D(i) = A; B = dec2binvec (A,8); c = bitxor(B6, B5, B4); C = circshift(B, [0 1]); C(1) = c; A = binvec2dec(C); end end
Algorithm 4. Pseudo-random sequences generation using LFSR for i = 1 to 256 D = LFSR(i); LFSROut(i, :) = D; end save output LFSROut
3.4. Rotation, permutation and XORing Here, the concept of the circular rotation, permutation and XOR using synthetic image (SImage1) is used. All row transformation in the right direction, and all column rotation up direction. All addition are done with mod(256).The image [W * H] elements are regarded as a set and are fed into an W * H matrix. Refer Algorithm 5. Algorithm 5. Rotation, Permutation and XORing for i = 1 to NR 1. XOR transformed data with synthetic image SImage1. 2. Permute each row coefficients using the pseudo-random sequence generated using LFSROut (Y1(X1(i + row number))). 3. Permute each column coefficients using the pseudo-random sequence generated using LFSROut (X1(Y1(i + column number))) 4. Transformed each row of the matrix using rotation function which is controlled by X1(Y1(i + row number)). 5. Transformed each column of the matrix using rotation function which is controlled by Y1(X1(i + column number)). end
The circulation functions are: Definition 1. The rotate Xir to rotate each data in the ith row of the matrix r times in the right direction. Definition 2. The rotate Yjs to rotate each data in the jth column of the matrix s times in the up direction. The calculation involved are easily explained using the following example
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0
1
B 9 B B B 17 B B 25 B Let matrix M ¼ B B 33 B B 41 B B @ 49 0
1
B 9 B B B 20 B B 25 B Rotate X 33 ¼ B B 33 B B 41 B B @ 49 57 0 1 B 9 B B B 17 B B 25 B 3 Rotate Y 3 ¼ B B 33 B B 41 B B @ 49
2
3
4
5
6
7
8
1
10 11 12 13 14 15 16 C C C 18 19 20 21 22 23 24 C C 26 27 28 29 30 31 32 C C C 34 35 36 37 38 39 40 C C 42 43 44 45 46 47 48 C C C 50 51 52 53 54 55 56 A
57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 C C C 21 22 23 24 17 18 19 C C 26 27 28 29 30 31 32 C C C 34 35 36 37 38 39 40 C C 42 43 44 45 46 47 48 C C C 50 51 52 53 54 55 56 A 58 59 60 61 62 63 64 1 2 27 4 5 6 7 8 10 35 12 13 14 15 16 C C C 18 43 20 21 22 23 24 C C 26 51 28 29 30 31 32 C C C 34 59 36 37 38 39 40 C C 42 3 44 45 46 47 48 C C C 50 11 52 53 54 55 56 A
57 58 19 60 61 62 63 64
3.5. Diffusion and XORing In cryptography, diffusion plays most important part, which refers to the process of spreading out the bits in the message so that redundancy in the plaintext is spread out over the complete cipher text. Especially in images, where the redundancy is large, the process of diffusion is a necessary requirement to develop a secure encryption technique. Diffusion process changes the statistical properties of the image by spreading the influence of each bit of the plain image all over the cipher image, which removes the possibility of differential attacks by comparing the pair of plain and cipher images. Refer Algorithm 6. Algorithm 6. Diffusion and XORing for i = 1 to ND 1. XOR transformed data with synthetic image SImage2. 2. Modify second row by XORing of the first row and second row, modify third row by XORing of the modified second row and third row and the process continues till the end of the image, which in turn gives the vertical diffusion. 3. Modify second last column by XORing of the last column and second last column, modify third last column by XORing of the modified second last column and third last column and the process continues till the first column, which in turn gives the horizontal diffusion. end
3.6. Decryption Decryption can be obtained as exact reverse of the encryption as discussed in above sections. 4. Security and performance analysis An robust encryption scheme should resist against various kinds of attacks such as ciphertext only attack, known plaintext attack, statistical attacks, brute-force attacks, etc. Results of the security and performance analysis performed on the proposed image encryption technique.
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600 500 400 300 200 100 0 50
100
150
200
250
0
50
100
150
200
250
Fig. 2. Experimental results: (a) Lena image; (b) encrypted image; (c) histogram of Lena image; (d) histogram of encrypted image.
4.1. Statistical analysis 4.1.1. Histograms of encrypted image We have analyzed of the histograms of 10 plain images and their corresponding cipher images using various combinations of the secret key. One example of such histogram analysis is shown in Fig. 2. We have taken the image Lena (256 * 256 pixels) and the secret keys (x0 = 4.94785238984676, y0 = 0.31256128342389, K = 131.7183545564243 and N = 110). It is clear that the histograms of the cipher image are fairly uniform and significantly different from the respective histograms of the plain image and hence does not provide any type of information to employ statistical attack on the encryption scheme. 4.1.2. Correlation of two adjacent pixels For any given image each pixel is highly correlated with its adjacent pixels in horizontal, vertical and diagonal direction. An ideal encryption technique should produce the encrypted image with no such correlation in the adjacent pixels. It is an effective tool to assess the measure of encryption-efficiency in terms minimum correlation between corresponding pixels. Let the size of image is W * H and represented as X(i, j) in matrix form.
P P PN N N N k¼1 fak bk g j¼1 ak k¼1 bk C ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n o P 2 P P PN 2 N N Nk¼1 fa2k g N Nk¼1 bk k¼1 fak g k¼1 fbk g
ð5Þ
1. Horizontal correlation ak = X(i, j) and bk = X(i, j + 1) with condition 1 6 i 6 H, 1 6 j 6 W 1 and take any N pairs of (a, b) samples with different values of i and j. 2. Vertical correlation ak = X(i, j) and bk = X(i + 1, j) with condition 1 6 i 6 H 1, 1 6 j 6 W and take any N pairs of (a, b) samples with different values of i and j. 3. Diagonal correlation ak = X(i, j) and bk = X(i + 1, j + 1) with condition 1 6 i 6 H 1, 1 6 j 6 W 1 and take any N pairs of (a, b) samples with different values of i and j. 4.1.3. Correlation between original and cipher images We have analyzed the correlation between various pairs of original and cipher images by computing the 2D correlation coefficients between original and encrypted images. We have chosen 10 images for this analysis (cf. Tables 1 and 2).
Table 1 Correlation coefficients for the two adjacent pixels in the original and encrypted shown in Fig. 2.
Horizontal Vertical Diagonal
Original image (Fig. 2(a))
Encrypted image (Fig. 2(b))
0.9187 0.9511 0.8934
0.0004992 0.00198 0.0008371
Table 2 Average correlation between the original and cipher images. Plain vs Encrypted image
Correlation 0.00182
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Table 3 Sensitivity analysis for the cipher to key. Test item
Test results
Sensitivity for cipher to key (key difference ( 10exp 14) in x0) Sensitivity for Cipher to Key (key difference ( 10exp 14) in y0)
0.00173 0.0005932
Fig. 3. Sensitivity analysis for the cipher to keys when encrypt with slightly different key: (a) encrypted image; (b) encrypted image with change in x; (c) encrypted image with change in y.
4.2. Sensitivity analysis of the encryption. 4.2.1. Sensitivity analysis for the cipher to key A good encryption algorithm must possess the property that a slight change in the key changes the encrypted file almost completely. The key sensitivity of a cryptosystem is observed by two different method. 1. Encrypted image should be very sensitive to the secret key. If we use two slightly different keys to encrypt the same image then the correlation between two encrypted images should be negligible. Results are shown in Table 3. (a) First Lena image is encrypted by using a pair of keys of 4.94785238984676 and 0.31256128342389, and then the least significant bit of key x is changed. The new keys become 4.94785238984675 and 0.31256128342389 in this example, which are used to encrypt the Lena image. The resulting image is Fig. 3(b). (b) Now change the least significant bit of y of the original key.The new keys become 4.94785238984676 and 0.31256128342388 in this example, which are used to encrypt the Lena image.The resulting image is Fig. 3(c). 2. The encrypted image should not be decrypted correctly if there is slight difference between the encryption and decryption keys. We will encrypt image by using the pairs of key 4.94785238984676 and 0.31256128342389. After try to decrypt with slight change in keys. First by changing least significant bit of x key, and decrypt with keys 4.94785238984675 and 0.31256128342389. The resulting image is Fig. 4(b). After that changing the least significant bit of y key, and decrypt with keys 4.94785238984676 and 0.31256128342388. The resulting image is Fig. 4(c). 4.2.2. Difference attacks sensitivity analysis for the cipher to plaintext For implementation of various types of attacks such as known plaintext attack, chosen plaintext attack and more advanced adaptive chosen plaintext attack, a slight change in plain image and compare the encrypted images in order to extract some relationship between plain image and encrypted image, which help attackers in determining the secret key. Such analysis is termed as differential cryptanalysis in cryptographic terminology. If one minor change in the original image causes large changes in the encrypted image then such differential analysis may become useless.
Fig. 4. Sensitivity analysis for the cipher to keys when decrypt with slightly different key: (a) decrypted image with original key; (b) decrypted Image with change in x; (c) decrypted Image with change in y.
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Test results
NPCR UACI
0.9972 0.32821
Fig. 5. Difference attacks sensitivity analysis for the cipher to plaintext: (a) encrypted image; (b) encrypted Image with 1-bit change in original image.
We have measured the number of pixels change rate by calculating the number of pixel change rate (NPCR) and the unified average changing intensity (UACI) of the two encrypted images. And we have also calculated the correlation between these two encrypted image which is 0.0002342.
PH PW NPCR ¼ UACI ¼
j¼1 fDij g 100 W" H # H W X X fC 1 ði; jÞ C 2 ði; jÞg 1 i¼1
W H
i¼1
j¼1
255
ð6Þ 100
ð7Þ
where W and H are the width and height of encrypted image. Two encrypted images C1 and C2, whose corresponding original images have only 1-pixel difference. We define a two-dimensional array D, which has the same size as C1 and C2. If C1(i, j) = C2(i, j), then Di,j = 0, otherwise Di,j = 1. The proposed image encryption technique shows extreme sensitivity on the original and hence not vulnerable to the differential attacks (see Table 4 and Fig. 5).
4.2.3. Avalanche criterion As we know that if one bit change in the plaintext should make ideally 50% difference in the bits of the cipher, and we also done analysis about the changing rate of the bits of the cipher, the changing rate is 49.97%, so our scheme is nearly ideal.
4.3. Experimental proof for chosen plaintext attack The various attacks are following that can be done by adversary. 1. Ciphertext only: Adversary possesses a string of cipher data. Adversary is to recover the plaintext, or find the key. 2. Known plaintext: Adversary possesses a string of plaintext a, and the corresponding cipher text b. Adversary is to find the key or an approach so that he can decrypt a new cipher encrypted with the help of the same key. 3. Chosen plaintext: Adversary has obtained temporary access to the encryption machinery. So, he can choose a plaintext of his choice string a, and can construct the corresponding encrypting string b. 4. Chosen cipher text: Adversary has obtained temporary access to the decryption machinery. So he can choose a ciphertext of his choice string b, and can get corresponding plaintext string a. Adversary is to find the key. Take picture X and Y are encrypted in the same key [3], and the plaintext Y as PlainY, the key as K, XOR key and Plaintext as 0 Z 0Y ¼ XORðPlainY; KÞ, the permutation and diffuse this of Z 0Y as Z Y ¼ PðZ 0Y Þ, and K = XOR(ZY, PlainY). 0 Permutation and diffusion performed on the XOR value Z 0Y , so it is not possible to decrypt Z 0X with the key K . Because 0 K – K. And therefore the attack matrix has no use in proposed technique. The original images are Figs. 6(a) and (b), the corresponding encrypted images are Figs. 6(c) and (d). We try to recover Fig. 6(b) from Fig. 6(a) and (c) by the iterating attack matrix, Fig. 6(e) on the basis of this we can say that the proposed scheme is safe to against the method of attack matrix. So the our scheme is safe to against the attack of iterating attack matrix [12].
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Fig. 6. Attack matrix: (a) Lena image; (b) house image; (c) encrypted Lena image; (d) encrypted house image; (e) attack to (b) by using (a) and (c).
4.4. Speed analysis The running speed of a cryptosystem is the most important parameter for the practical applications. In order to achieve this,the computational complexity is decreased by decreasing the numbers of multiplication and floating-points number to integer conversion [13]. To support our analysis, we have performed various basic operations which are repeated 10,000,000 times on two computers with different configuration, where PC1 with a 2.1 GHz Pentium processor and 2G RAM and PC2 with a 2.67 GHz HPZ400 workstation and 6G RAM. Results are shown in Table 6 shows that multiplication and conversion from floating-points to integers should be avoided to get high efficiency for encryption and decryption. Let consider the gray scale image of W width and H height (Table 5). 4.4.1. Patidar cryptosystem 1. Calculation of XOR key (a) Multiplication: ’2 * W * H. (b) Floating-points number to integer conversion: ’W * H. (c) Modolus: ’0.5 * W * H. 2. Confusion using XOR key (a) XOR: ’W * H. 3. Diffusion (a) XOR: ’2 * W * H. 4. Generation of CKS (a) Total multiplication: ’7 * W * H. (b) Floating-points number to integer conversion: ’W * H. (c) Modulus: ’2 * W * H. (d) Addition: ’3 * W * H. 5. Confusion using CKS image (a) XOR: ’W * H. 6. Total (a) Multiplication: ’9 * W * H. (b) Floating-points number to integer conversion: ’2 * W * H. (c) XOR: ’4 * W * H. (d) Modulus: ’3 * W * H. (e) Addition: ’3 * W * H. 4.4.2. Proposed cryptosystem 1. Generation of Synthetic image (a) Multiplication: ’6 * 256. (b) Floating-points number to integer conversion: ’ 6 * 256. Table 5 Computational complexity. Computation
Patidar cryptosystem
Proposed cryptosystem
FP number to integer conversion Multiplication Modulus XOR Addition Permutation Rotation
2*W*H 9*W*H 3*W*H 4*W*H 3*W*H – –
1536 1536 1536 (3 * ND + NR + 2) * W * H – NR * (W + H) NR * (W + H)
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Table 6 Time required for various basic operations (ms).
2.
3.
4. 5.
Operators
PC1
PC2
Floating point number to integer conversion Multiplication Modulus XOR Addition
190 60 15 15 18
145 42 10 10 12
(c) Modolus: ’2 * 256. (d) XOR: ’2 * W * H. Numbers of Rounds (a) XOR: ’3 * W * H. (b) Modolus: ’2. Rotation (a) XOR: ’NR * W * H. (b) Permutation: ’NR * (W + H). (c) Tranformation: ’NR * (W + H). Diffusion (a) XOR: ’3 * ND * W * H. Total (a) Multiplication: ’6 * 256. (b) Floating-points number to integer conversion: ’ 6 * 256. (c) XOR: ’(3 * ND + NR + 2) * W * H. (d) Modulus: ’6 * 256. (e) Permutation: ’NR * (W + H). (f) Rotation: ’NR * (W + H).
4.5. Entropy Entropy of a message m measured by
HðMÞ ¼
N X
pðmi Þ log
i¼1
1 pðmi Þ
ð8Þ
where N is the total number of symbols, p(mi) represents the probability of occurrence of symbol mi. For an ideal random image of 256 symbols entropy is equal to 8-bits. For encrypted images 6(c), (d) entropies are 7.9996 and 7.9993, respectively. This shows that encrypted images are very close to the random image. Hence, the proposed method is secure against the entropy attack. 4.6. Key space analysis The key space should be large enough. The effective key space for the proposed method is of nearly 157-bits [1]. This is large enough to resist the brute-force attack. 5. Conclusions The improved encryption concept is proposed using the rotation, diffusion and numbers of rounds for both stages depends upon plain image and key combination. This method is immune to various types of cryptographic attacks like known plaintext, chosen plaintext attacks. The proposed algorithm is lossless. The proposed scheme is secure and useful for realtime application. Acknowledgement This work is part of the Research Project funded by All India Council of Technical Education (Government of India) vide their office order: F.No. 8023/BOR/RID/RPS-236/2008-09. Also thankful to Department of Science & Technology (Ministry of Science & Technology, Government of India) vide Ref. No. SR/SS/963/2009-10 under Research and Development Support SERC (1009) Scheme.
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References [1] Patidar V, Pareek NK, Sud KK. A new substitution diffusion based image cipher using chaotic standard and logistic maps. Commun Nonlinear Sci Numer Simulat 2009;14:3056–75. [2] Schneier B. Applied cryptography: protocols algorithms and source code in C. New York (USA): Wiley; 1996. [3] Tong Xiaojun, Cui Minggen. Image encryption with compound chaotic sequence cipher shifting dynamically. Image Vision Comput 2008;26:843–50. [4] Mazloom Sahar, Masud Eftekhari-Moghadam Amir. Color image encryption based on coupled nonlinear chaotic map. Chaos Solitons Fract 2009;42:1745–54. [5] Chen HC, Guo JI, Huang LC, Yen JC. Design and realization of a new signal security system for multimedia data transmission. EURASIP J Appl Signal Process 2003;2003(13):1291–305. [6] Wang Yong, Wong Kwok-Wo, Liao Xiaofeng, Xiang Tao, Chen Guanrong. A chaos-based image encryption algorithm with variable control parameters chaos. Solitons Fract 2009;41:1773–83. [7] Li Chengqing, Li Shujun, Chen Guanrong, Halang Wolfgang A. Cryptanalysis of an image encryption scheme based on a compound chaotic sequence. Image Vision Comput 2009;27:1035–9. [8] Rhouma Rhouma, Ercan Solak, Safya Belghith. Cryptanalysis of a new substitution–diffusion based image cipher. Commun Nonlinear Sci Numer Simulat. doi:10.1016/j.cnsns.2009.07.007. [9] Alvarez G, Li Shujun. Cryptanalyzing a nonlinear chaotic algorithm (NCA) for image encryption. Commun Nonlinear Sci Numer Simulat 2009;14:3743–9. [10] Lian S, Sun J, Wang Z. A block cipher based on a suitable use of chaotic standard map. Chaos Solitons Fract 2005;26:117–29. [11] Rajpal N, Kumar A, Jindal PR. Demonstrating the use of error coding technique in the field of steganography, along with linear feedback shift register technique. In: Proc of the 2nd Workshop on Computer Vision, Graphics and Image Processing, Gwalior, India; 2004. p. 22–7. [12] Tong Xiaojun, Cui Minggen, Wang Zhu. A new feedback image encryption scheme based on perturbation with dynamical compound chaotic sequence cipher generator. Opt Commun 2009;282:2722–8. [13] Yong Wang, Kwok-Wo Wong, Xiaofeng Liao, Guanrong Chen. A new chaos-based fast image encryption algorithm. Appl Soft Comput J. doi:10.1016/ j.asoc.2009.12.011.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Extended substitution–diffusion based image cipher using chaotic standard map Anil Kumar *, M.K. Ghose Computer Science and Engineering Department, Sikkim Manipal Institute of Technology, Sikkim, India
a r t i c l e
i n f o
Article history: Received 25 September 2009 Received in revised form 28 January 2010 Accepted 12 April 2010 Available online 20 April 2010 Keywords: Chaotic standard map Diffusion Encryption
a b s t r a c t This paper proposes an extended substitution–diffusion based image cipher using chaotic standard map [1] and linear feedback shift register to overcome the weakness of previous technique by adding nonlinearity. The first stage consists of row and column rotation and permutation which is controlled by the pseudo-random sequences which is generated by standard chaotic map and linear feedback shift register, second stage further diffusion and confusion is obtained in the horizontal and vertical pixels by mixing the properties of the horizontally and vertically adjacent pixels, respectively, with the help of chaotic standard map. The number of rounds in both stage are controlled by combination of pseudo-random sequence and original image. The performance is evaluated from various types of analysis such as entropy analysis, difference analysis, statistical analysis, key sensitivity analysis, key space analysis and speed analysis. The experimental results illustrate that performance of this is highly secured and fast. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Security of multimedia data is most important days due to the widespread transmission over various communication networks. It has been observed that the traditional encryption schemes [2] fail to protect multimedia data due to some special properties and some specific requirements of multimedia processing systems, such as mammoth size and strong redundancy of uncompressed data. The fundamental features of chaotic systems such as ergodicity, mixing property, sensitivity to initial conditions, system parameters and which can be considered analogous to ideal cryptographic properties such as confusion, diffusion, balance and avalanche properties. Hence, many chaos-based encryption systems also proposed [1,3–6]. Many chaos-based cryptography schemes have been successfully cryptanalyzed [7–9]. In this paper, we have proposed an extended symmetric image encryption by using the chaotic 2D standard map and linear feedback shift register which in turn introduces the nonlinearity which is the main limitation of the Patidar et al. [1]. The proposed algorithm comprises three stages, viz. (i) Generation of pseudo-random sequences and synthetic images. (ii) Horizontal, vertical rotation and permutation and XORing with synthetic image. (iii) Vertical, horizontal diffusion and XORing with synthetic image. Pseudo-random sequences is generated by linear feedback shift register and standard chaotic map, while the synthetic images are generated using standard chaotic map. Total number of rounds for two stages are calculated using pseudo-random sequence which is generated using standard chaotic map and original image. * Corresponding author. E-mail address: [email protected] (A. Kumar). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.04.010
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The rest of the paper is organized as follows: In Section 2, brief description chaos, linear feedback shift register and of the Patidar [1] cryptosystem. The detailed algorithms of the proposed encryption procedures is discussed in Section 3. In Section 4, analysis of the proposed image cipher and evaluate its performance through various statistical analysis, key sensitivity analysis, differential analysis, key space analysis, speed analysis, etc. Finally, Section 5 concludes the paper.
2. Basic of chaos, LFSR and brief discussion of cryptosystem [1] 2.1. Chaotic sequence It has a long circle, sensitive to initial value, and unpredictability. The encryption with chaos is fast, so it is widely combined with traditional encryption. The general chaotic system model is given as below
xðnÞ ¼ f ðxðn 1ÞÞ
ð1Þ
where the x(n) is a chaotic sequence generated by the nonlinear f(.), x(0) are the initial condition values. The 1D logistic maps whose chaotic intervals are both in [0, 1] as follows
X nþ1 ¼ X n lð1 X n Þ
ð2Þ
where l[0, 4] and X[0, 1] The compound chaotic functions whose chaotic intervals are both in [1, 1] as follows
( FðxÞ ¼
8x4 8x2 þ 1 . . . ; 4x3 3x . . . ;
x<0
x P 0:
ð3Þ
The chaotic standard map have a large key space compared to another maps Lian et al. [10].
X nþ1 ¼ X n þ K sinY n
ð4Þ
Y nþ1 ¼ Y n þ X nþ1
where Xn and Yn are taken modulo 2p and K is constant. We will generate the pseudo-random sequences using Eq. (4) which in turn used for generating the synthetic images and in calculating the numbers of rounds.
2.2. Linear feedback shift register A Linear feedback shift register (LFSR) is a mechanism for generating binary sequences [11]. Fig. 1 shows a general model of an n-bit LFSR. LFSR are extremely good pseudo-random binary sequence generators [11]. When this register is loaded with any given initial value (except 0 which will generate a pseudo-random binary sequence of all 0s) it generates pseudo-random binary sequence which has very good randomness and statistical properties. The only signal necessary for the generation of the binary sequence is a clock pulse. With each clock pulse a bit of the binary sequence is produced. A example of 4-bit LFSR is considered to demonstrate the functioning of LFSR with the feedback function f = 1 + x + x4. Its initial bit values are used (1111). The output sequence Zn: 011111000000001. . . Generated by LFSR in is periodic of period 15. Period of the sequence generated by LFSR is maximum if we use the primitive polynomial. To design any stream cipher system, one needs to consider the LFSR with primitive feedback polynomials as the basic building blocks. The pseudo-random sequence depends upon the initial seed value and feedback function.
Fig. 1. A general model of n bit linear feedback shift register.
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2.3. Brief description of cryptosystem [1] Patidar et al. [1] proposed a cryptosystem which was utilizes the chaotic 2D standard map and 1D logistic map and which is specifically designed for the coloured images. The initial condition value, number of iterations together consider as the secret key for the algorithm. The algorithm comprises four rounds: two for the substitution and two for the diffusion. The first round of substitution/confusion is achieved with the help of intermediate XORing keys calculated from the secret key. Then two rounds of diffusion namely the horizontal and vertical diffusions are completed by mixing the properties of horizontally and vertically adjacent pixels, respectively. In the fourth round, a robust substitution/confusion is accomplished by generating an intermediate chaotic key stream (CKS) image in a novel manner with the help of chaotic standard and logistic maps. Rhouma et al. [8] shows that all the steps are linear in nature and Rhouma also shows that this cryptosystem vulnerable to attacks by dividing the cryptosystem procedure in two majors successive steps: (1) the diffusion process and (2) the masking process. 3. Proposed method In the encryption, it includes the input image, secret keys. First, generation of pseudo-random sequences and synthetic images using standard chaotic map explained in Section 3.1, calculation of the numbers of rounds in Section 3.2, generation of pseudo-random sequence using LFSR explained in Section 3.3, rotation and permutation and XORing with synthetic image explained in Section 3.4, diffusion and XORing with synthetic image explained in Section 3.5, and decryption discussed in Section 3.6. 3.1. Generation of synthetic images and pseudo-random sequence using standard chaotic map Here, generation of synthetic images and pseudo-random sequences is discussed. Refer Algorithm 1. As we know that conversion floating-point number to integer is required for the chaos based encryption, this conversion and multiplication are time consuming compare to modulus, addition, XOR, etc. operations [13]. Hence, in the proposed method for generation of the synthetic images floating-points to integer conversions and multiplication is reduced into order decrease the computational complexity. 1. The secret key consists of three floating-point numbers and one integer (x0, y0, K, N), where x0, y0 2 (0, 2p), K can have any real value greater than 18.0 and N is any integer value, ideally should be greater than 100. 2. By iterating Eq. (4) W * H + N times, the pseudo-random sequence generated as XKey and YKey by taking last 256 values. 3. The synthetic images and scaled pseudo-random sequence are explained in Algorithm 1.
Algorithm 1. Synthetic images generation and scaled pseudorandom sequence for i = 1 to 256 j k X1ðiÞ ¼ XKeyðiÞ 256 ; 2p j k Y1ðiÞ ¼ YKeyðiÞ 256 ; 2p end for i = 257 to W * H X1ðiÞ ¼ Y1ðX1ði 256ÞÞ bsðY1ðX1ði 128ÞÞÞ bsðY1ðX1ði 64ÞÞÞ; //Where bs is the byte substitution taken from AES Y1ðiÞ ¼ XKey1ðY1ði 256ÞÞ bsðX1ðY1ði 128ÞÞÞ bsðX1ðY1ði 64ÞÞÞ; end SImage1 = reshape (X1,W,H); SImage2 = reshape (Y1,W,H); 3.2. Numbers of rounds The numbers of rounds for both rotation and diffusion is depends upon the pseudo-random sequence generated using standard chaotic map and original image. Refer Algorithm 2. Algorithm 2. Numbers of rounds 1. 2. 3. 4.
Read the image, let the size of image is W * H. XOR all the bytes of the image let it be IXOR. XOR of the values of the pseudo-random sequence X1 as KXORX and of Y1 as KXORY. The number of the rounds for rotation NR = mod(XOR(IXOR, KXORX), 5) + 1. The number of the rounds for diffusion ND = mod(XOR(IXOR, KXORY), 5) + 1.
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3.3. Generation of pseudo-random sequence using LFSR Here, the pseudo-random sequence generated using LFSR for all values (1, 256) considering the size of image is 256 256. Considering the original image size is 256 256. Generate the pseudo-random sequences using the primitive root x8 + x6 + x5 + x4 + 1. Refer Algorithms 3 and 4. Algorithm 3. Function to calculate pseudo-random sequence generation using LFSR function D = LFSR (input) for i = 1 to 256 if i = 256 D(i) = 256; else D(i) = A; B = dec2binvec (A,8); c = bitxor(B6, B5, B4); C = circshift(B, [0 1]); C(1) = c; A = binvec2dec(C); end end
Algorithm 4. Pseudo-random sequences generation using LFSR for i = 1 to 256 D = LFSR(i); LFSROut(i, :) = D; end save output LFSROut
3.4. Rotation, permutation and XORing Here, the concept of the circular rotation, permutation and XOR using synthetic image (SImage1) is used. All row transformation in the right direction, and all column rotation up direction. All addition are done with mod(256).The image [W * H] elements are regarded as a set and are fed into an W * H matrix. Refer Algorithm 5. Algorithm 5. Rotation, Permutation and XORing for i = 1 to NR 1. XOR transformed data with synthetic image SImage1. 2. Permute each row coefficients using the pseudo-random sequence generated using LFSROut (Y1(X1(i + row number))). 3. Permute each column coefficients using the pseudo-random sequence generated using LFSROut (X1(Y1(i + column number))) 4. Transformed each row of the matrix using rotation function which is controlled by X1(Y1(i + row number)). 5. Transformed each column of the matrix using rotation function which is controlled by Y1(X1(i + column number)). end
The circulation functions are: Definition 1. The rotate Xir to rotate each data in the ith row of the matrix r times in the right direction. Definition 2. The rotate Yjs to rotate each data in the jth column of the matrix s times in the up direction. The calculation involved are easily explained using the following example
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0
1
B 9 B B B 17 B B 25 B Let matrix M ¼ B B 33 B B 41 B B @ 49 0
1
B 9 B B B 20 B B 25 B Rotate X 33 ¼ B B 33 B B 41 B B @ 49 57 0 1 B 9 B B B 17 B B 25 B 3 Rotate Y 3 ¼ B B 33 B B 41 B B @ 49
2
3
4
5
6
7
8
1
10 11 12 13 14 15 16 C C C 18 19 20 21 22 23 24 C C 26 27 28 29 30 31 32 C C C 34 35 36 37 38 39 40 C C 42 43 44 45 46 47 48 C C C 50 51 52 53 54 55 56 A
57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 C C C 21 22 23 24 17 18 19 C C 26 27 28 29 30 31 32 C C C 34 35 36 37 38 39 40 C C 42 43 44 45 46 47 48 C C C 50 51 52 53 54 55 56 A 58 59 60 61 62 63 64 1 2 27 4 5 6 7 8 10 35 12 13 14 15 16 C C C 18 43 20 21 22 23 24 C C 26 51 28 29 30 31 32 C C C 34 59 36 37 38 39 40 C C 42 3 44 45 46 47 48 C C C 50 11 52 53 54 55 56 A
57 58 19 60 61 62 63 64
3.5. Diffusion and XORing In cryptography, diffusion plays most important part, which refers to the process of spreading out the bits in the message so that redundancy in the plaintext is spread out over the complete cipher text. Especially in images, where the redundancy is large, the process of diffusion is a necessary requirement to develop a secure encryption technique. Diffusion process changes the statistical properties of the image by spreading the influence of each bit of the plain image all over the cipher image, which removes the possibility of differential attacks by comparing the pair of plain and cipher images. Refer Algorithm 6. Algorithm 6. Diffusion and XORing for i = 1 to ND 1. XOR transformed data with synthetic image SImage2. 2. Modify second row by XORing of the first row and second row, modify third row by XORing of the modified second row and third row and the process continues till the end of the image, which in turn gives the vertical diffusion. 3. Modify second last column by XORing of the last column and second last column, modify third last column by XORing of the modified second last column and third last column and the process continues till the first column, which in turn gives the horizontal diffusion. end
3.6. Decryption Decryption can be obtained as exact reverse of the encryption as discussed in above sections. 4. Security and performance analysis An robust encryption scheme should resist against various kinds of attacks such as ciphertext only attack, known plaintext attack, statistical attacks, brute-force attacks, etc. Results of the security and performance analysis performed on the proposed image encryption technique.
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600 500 400 300 200 100 0 50
100
150
200
250
0
50
100
150
200
250
Fig. 2. Experimental results: (a) Lena image; (b) encrypted image; (c) histogram of Lena image; (d) histogram of encrypted image.
4.1. Statistical analysis 4.1.1. Histograms of encrypted image We have analyzed of the histograms of 10 plain images and their corresponding cipher images using various combinations of the secret key. One example of such histogram analysis is shown in Fig. 2. We have taken the image Lena (256 * 256 pixels) and the secret keys (x0 = 4.94785238984676, y0 = 0.31256128342389, K = 131.7183545564243 and N = 110). It is clear that the histograms of the cipher image are fairly uniform and significantly different from the respective histograms of the plain image and hence does not provide any type of information to employ statistical attack on the encryption scheme. 4.1.2. Correlation of two adjacent pixels For any given image each pixel is highly correlated with its adjacent pixels in horizontal, vertical and diagonal direction. An ideal encryption technique should produce the encrypted image with no such correlation in the adjacent pixels. It is an effective tool to assess the measure of encryption-efficiency in terms minimum correlation between corresponding pixels. Let the size of image is W * H and represented as X(i, j) in matrix form.
P P PN N N N k¼1 fak bk g j¼1 ak k¼1 bk C ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n o P 2 P P PN 2 N N Nk¼1 fa2k g N Nk¼1 bk k¼1 fak g k¼1 fbk g
ð5Þ
1. Horizontal correlation ak = X(i, j) and bk = X(i, j + 1) with condition 1 6 i 6 H, 1 6 j 6 W 1 and take any N pairs of (a, b) samples with different values of i and j. 2. Vertical correlation ak = X(i, j) and bk = X(i + 1, j) with condition 1 6 i 6 H 1, 1 6 j 6 W and take any N pairs of (a, b) samples with different values of i and j. 3. Diagonal correlation ak = X(i, j) and bk = X(i + 1, j + 1) with condition 1 6 i 6 H 1, 1 6 j 6 W 1 and take any N pairs of (a, b) samples with different values of i and j. 4.1.3. Correlation between original and cipher images We have analyzed the correlation between various pairs of original and cipher images by computing the 2D correlation coefficients between original and encrypted images. We have chosen 10 images for this analysis (cf. Tables 1 and 2).
Table 1 Correlation coefficients for the two adjacent pixels in the original and encrypted shown in Fig. 2.
Horizontal Vertical Diagonal
Original image (Fig. 2(a))
Encrypted image (Fig. 2(b))
0.9187 0.9511 0.8934
0.0004992 0.00198 0.0008371
Table 2 Average correlation between the original and cipher images. Plain vs Encrypted image
Correlation 0.00182
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Table 3 Sensitivity analysis for the cipher to key. Test item
Test results
Sensitivity for cipher to key (key difference ( 10exp 14) in x0) Sensitivity for Cipher to Key (key difference ( 10exp 14) in y0)
0.00173 0.0005932
Fig. 3. Sensitivity analysis for the cipher to keys when encrypt with slightly different key: (a) encrypted image; (b) encrypted image with change in x; (c) encrypted image with change in y.
4.2. Sensitivity analysis of the encryption. 4.2.1. Sensitivity analysis for the cipher to key A good encryption algorithm must possess the property that a slight change in the key changes the encrypted file almost completely. The key sensitivity of a cryptosystem is observed by two different method. 1. Encrypted image should be very sensitive to the secret key. If we use two slightly different keys to encrypt the same image then the correlation between two encrypted images should be negligible. Results are shown in Table 3. (a) First Lena image is encrypted by using a pair of keys of 4.94785238984676 and 0.31256128342389, and then the least significant bit of key x is changed. The new keys become 4.94785238984675 and 0.31256128342389 in this example, which are used to encrypt the Lena image. The resulting image is Fig. 3(b). (b) Now change the least significant bit of y of the original key.The new keys become 4.94785238984676 and 0.31256128342388 in this example, which are used to encrypt the Lena image.The resulting image is Fig. 3(c). 2. The encrypted image should not be decrypted correctly if there is slight difference between the encryption and decryption keys. We will encrypt image by using the pairs of key 4.94785238984676 and 0.31256128342389. After try to decrypt with slight change in keys. First by changing least significant bit of x key, and decrypt with keys 4.94785238984675 and 0.31256128342389. The resulting image is Fig. 4(b). After that changing the least significant bit of y key, and decrypt with keys 4.94785238984676 and 0.31256128342388. The resulting image is Fig. 4(c). 4.2.2. Difference attacks sensitivity analysis for the cipher to plaintext For implementation of various types of attacks such as known plaintext attack, chosen plaintext attack and more advanced adaptive chosen plaintext attack, a slight change in plain image and compare the encrypted images in order to extract some relationship between plain image and encrypted image, which help attackers in determining the secret key. Such analysis is termed as differential cryptanalysis in cryptographic terminology. If one minor change in the original image causes large changes in the encrypted image then such differential analysis may become useless.
Fig. 4. Sensitivity analysis for the cipher to keys when decrypt with slightly different key: (a) decrypted image with original key; (b) decrypted Image with change in x; (c) decrypted Image with change in y.
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Test results
NPCR UACI
0.9972 0.32821
Fig. 5. Difference attacks sensitivity analysis for the cipher to plaintext: (a) encrypted image; (b) encrypted Image with 1-bit change in original image.
We have measured the number of pixels change rate by calculating the number of pixel change rate (NPCR) and the unified average changing intensity (UACI) of the two encrypted images. And we have also calculated the correlation between these two encrypted image which is 0.0002342.
PH PW NPCR ¼ UACI ¼
j¼1 fDij g 100 W" H # H W X X fC 1 ði; jÞ C 2 ði; jÞg 1 i¼1
W H
i¼1
j¼1
255
ð6Þ 100
ð7Þ
where W and H are the width and height of encrypted image. Two encrypted images C1 and C2, whose corresponding original images have only 1-pixel difference. We define a two-dimensional array D, which has the same size as C1 and C2. If C1(i, j) = C2(i, j), then Di,j = 0, otherwise Di,j = 1. The proposed image encryption technique shows extreme sensitivity on the original and hence not vulnerable to the differential attacks (see Table 4 and Fig. 5).
4.2.3. Avalanche criterion As we know that if one bit change in the plaintext should make ideally 50% difference in the bits of the cipher, and we also done analysis about the changing rate of the bits of the cipher, the changing rate is 49.97%, so our scheme is nearly ideal.
4.3. Experimental proof for chosen plaintext attack The various attacks are following that can be done by adversary. 1. Ciphertext only: Adversary possesses a string of cipher data. Adversary is to recover the plaintext, or find the key. 2. Known plaintext: Adversary possesses a string of plaintext a, and the corresponding cipher text b. Adversary is to find the key or an approach so that he can decrypt a new cipher encrypted with the help of the same key. 3. Chosen plaintext: Adversary has obtained temporary access to the encryption machinery. So, he can choose a plaintext of his choice string a, and can construct the corresponding encrypting string b. 4. Chosen cipher text: Adversary has obtained temporary access to the decryption machinery. So he can choose a ciphertext of his choice string b, and can get corresponding plaintext string a. Adversary is to find the key. Take picture X and Y are encrypted in the same key [3], and the plaintext Y as PlainY, the key as K, XOR key and Plaintext as 0 Z 0Y ¼ XORðPlainY; KÞ, the permutation and diffuse this of Z 0Y as Z Y ¼ PðZ 0Y Þ, and K = XOR(ZY, PlainY). 0 Permutation and diffusion performed on the XOR value Z 0Y , so it is not possible to decrypt Z 0X with the key K . Because 0 K – K. And therefore the attack matrix has no use in proposed technique. The original images are Figs. 6(a) and (b), the corresponding encrypted images are Figs. 6(c) and (d). We try to recover Fig. 6(b) from Fig. 6(a) and (c) by the iterating attack matrix, Fig. 6(e) on the basis of this we can say that the proposed scheme is safe to against the method of attack matrix. So the our scheme is safe to against the attack of iterating attack matrix [12].
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Fig. 6. Attack matrix: (a) Lena image; (b) house image; (c) encrypted Lena image; (d) encrypted house image; (e) attack to (b) by using (a) and (c).
4.4. Speed analysis The running speed of a cryptosystem is the most important parameter for the practical applications. In order to achieve this,the computational complexity is decreased by decreasing the numbers of multiplication and floating-points number to integer conversion [13]. To support our analysis, we have performed various basic operations which are repeated 10,000,000 times on two computers with different configuration, where PC1 with a 2.1 GHz Pentium processor and 2G RAM and PC2 with a 2.67 GHz HPZ400 workstation and 6G RAM. Results are shown in Table 6 shows that multiplication and conversion from floating-points to integers should be avoided to get high efficiency for encryption and decryption. Let consider the gray scale image of W width and H height (Table 5). 4.4.1. Patidar cryptosystem 1. Calculation of XOR key (a) Multiplication: ’2 * W * H. (b) Floating-points number to integer conversion: ’W * H. (c) Modolus: ’0.5 * W * H. 2. Confusion using XOR key (a) XOR: ’W * H. 3. Diffusion (a) XOR: ’2 * W * H. 4. Generation of CKS (a) Total multiplication: ’7 * W * H. (b) Floating-points number to integer conversion: ’W * H. (c) Modulus: ’2 * W * H. (d) Addition: ’3 * W * H. 5. Confusion using CKS image (a) XOR: ’W * H. 6. Total (a) Multiplication: ’9 * W * H. (b) Floating-points number to integer conversion: ’2 * W * H. (c) XOR: ’4 * W * H. (d) Modulus: ’3 * W * H. (e) Addition: ’3 * W * H. 4.4.2. Proposed cryptosystem 1. Generation of Synthetic image (a) Multiplication: ’6 * 256. (b) Floating-points number to integer conversion: ’ 6 * 256. Table 5 Computational complexity. Computation
Patidar cryptosystem
Proposed cryptosystem
FP number to integer conversion Multiplication Modulus XOR Addition Permutation Rotation
2*W*H 9*W*H 3*W*H 4*W*H 3*W*H – –
1536 1536 1536 (3 * ND + NR + 2) * W * H – NR * (W + H) NR * (W + H)
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Table 6 Time required for various basic operations (ms).
2.
3.
4. 5.
Operators
PC1
PC2
Floating point number to integer conversion Multiplication Modulus XOR Addition
190 60 15 15 18
145 42 10 10 12
(c) Modolus: ’2 * 256. (d) XOR: ’2 * W * H. Numbers of Rounds (a) XOR: ’3 * W * H. (b) Modolus: ’2. Rotation (a) XOR: ’NR * W * H. (b) Permutation: ’NR * (W + H). (c) Tranformation: ’NR * (W + H). Diffusion (a) XOR: ’3 * ND * W * H. Total (a) Multiplication: ’6 * 256. (b) Floating-points number to integer conversion: ’ 6 * 256. (c) XOR: ’(3 * ND + NR + 2) * W * H. (d) Modulus: ’6 * 256. (e) Permutation: ’NR * (W + H). (f) Rotation: ’NR * (W + H).
4.5. Entropy Entropy of a message m measured by
HðMÞ ¼
N X
pðmi Þ log
i¼1
1 pðmi Þ
ð8Þ
where N is the total number of symbols, p(mi) represents the probability of occurrence of symbol mi. For an ideal random image of 256 symbols entropy is equal to 8-bits. For encrypted images 6(c), (d) entropies are 7.9996 and 7.9993, respectively. This shows that encrypted images are very close to the random image. Hence, the proposed method is secure against the entropy attack. 4.6. Key space analysis The key space should be large enough. The effective key space for the proposed method is of nearly 157-bits [1]. This is large enough to resist the brute-force attack. 5. Conclusions The improved encryption concept is proposed using the rotation, diffusion and numbers of rounds for both stages depends upon plain image and key combination. This method is immune to various types of cryptographic attacks like known plaintext, chosen plaintext attacks. The proposed algorithm is lossless. The proposed scheme is secure and useful for realtime application. Acknowledgement This work is part of the Research Project funded by All India Council of Technical Education (Government of India) vide their office order: F.No. 8023/BOR/RID/RPS-236/2008-09. Also thankful to Department of Science & Technology (Ministry of Science & Technology, Government of India) vide Ref. No. SR/SS/963/2009-10 under Research and Development Support SERC (1009) Scheme.
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