Advances in Engineering Software 49 (2012) 29–42
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Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Novel Approach for fast Compressed Hybrid color image Cryptosystem Kamlesh Gupta a,⇑, Sanjay Silakari b a b
Dept. of Computer Science & Eng., Jaypee University of Engineering and Technology, A.B. Road, Guna, MP, India Dept. of Computer Science & Eng., University Institute of Technology, RGPV, Bhopal, MP, India
a r t i c l e
i n f o
Article history: Received 25 January 2012 Received in revised form 29 February 2012 Accepted 4 March 2012 Available online 25 April 2012 Keywords: Chaotic map Cat map Standard map Elliptic Curve Cryptography Curvelet transform ECDLP
a b s t r a c t In this Paper, the issues pertaining with efficient, fast, cost effective and secured image transmission are addressed in totality. The proposed model employs Compressed Hybrid Cryptosystem constitutes compression, encryption and secured session key exchange along with the transmission of image. In the proposed work, an algorithm has been designed to generate diffusion template using 3D Standard map. The image is rotated vertically and horizontally followed by a shuffle using 3D Cat map and Standard map. The image is then encrypted by performing XOR operation on the shuffled image and diffusion template. Proposed method takes lesser time and is found to be safe from any of the existing cryptanalytic attack. Further Elliptic Curve Cryptography is used for secure transfer of private key, which has resulted in significant reduction in the key size without compromising its security strength. To reduce bandwidth requirement and power consumption, a compression technique is proposed based on curvelet transform before image encryption, with special technique of coefficient elimination by which a higher compression ratio can be obtained without much loss in image information. Even though the coefficients neglected are large, the higher PSNR values show that curvelet has better reconstruction performance. The model has been rigorously examined over the prevalent standard test and has encouragingly succeeded to pass most of them like key sensitivity analysis, key space analysis, statistical analysis, differential analysis, entropy analysis, randomness analysis, PSNR analysis, MSE analysis, for fast, cost effective and secured image transmission. Which was the key problem statement for this research work. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction With the advent of Internet and World Wide Web, the amount of digital information to be stored and communicated has grown exponentially beyond imagination. This digital information not only comprises text, but also has large volume of image, audio/video and multimedia data, which comparatively is very bulky than the textual information. The images as on date have become an integral and vital component of any useful data and are widely used in several important applications. Few of these crucial applications include Military Image Database & Message Communication, Confidential Video Conferencing, Medical Imaging System & Telemedicine, Online Personal Photograph Albums, Natural Disaster or Catastrophe Alarming Systems, Online Image Identification and Authentication, Reflection Seismology, Electronic Surveillance Systems, Document Imaging, Image ‘CAPTCHA’, Image Registration, Geographic Information System, etc. ⇑ Corresponding author. Tel.: +91 94257 57684. E-mail addresses:
[email protected] (K. Gupta),
[email protected] (S. Silakari). 0965-9978/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advengsoft.2012.03.001
The use of images in most these applications has given rise to several problems as follows: 1. Traffic on the internet has increased tremendously resulting in longer delays and higher communication cost. 2. An image being bulky amount of data requires larger bandwidth for the transmission. 3. High space requirements for intermediate and final storage. 4. Security and integrity of the transmitted images. All these applications not only require faster communication but also require that image transmission must be cost effective and secured. Encryption of images is different from that of textual data, as images are intrinsically bulky and have high correlation among pixels and higher redundancy which is difficult to be handled by the traditional encryption schemes. Hence the DES, AES, IDEA, Blowfish, RC6 and RSA, etc., do not suite for modern image transmission requirements. Many researchers have tried to innovate better solutions for secured image transmission. In particular, application of chaos theory in multimedia encryption is one of the important research directions.
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The aim of this research is to fix the following problems for efficient, fast, cost effective and secured image transmission. Asymmetric cryptography doesn’t suite for secured transmission of images because of the bulk data, strong pixel correlation and high redundancy. Moreover encryption at the source and decryption at the destination lowers the encryption performance. Symmetric encryption is fit for image encryption, but the security of symmetric encryption depends on the private key so it is needed to transmit this key by asymmetric method but the key used in that is itself bulky, hence, it is needed to transfer the key by secured channel with a significant reduction in key size. An image is the bulky amount of data and requires larger bandwidth for the transmission, hence efficient compression is required. To view the basic ingredients, speed, cost effectiveness and security of image transmission in totality. In this paper, we propose the following solutions to the above mentioned problems by proposed Compressed Hybrid Cryptosystem, which are described below: The algorithm has been designed to generate diffusion template using 3D standard map. The image is rotated vertically and horizontally followed by a shuffle using 3D cat map and standard map. The image is then encrypted by performing XOR operation on the shuffled image and diffusion template. Proposed method takes lesser time and is found to be safe from any of the existing cryptanalytic attack. Further Elliptic Curve Cryptography is used for secure transfer of private key, which has resulted in significant reduction in the key size without compromising its security strength. To reduce bandwidth requirement and power consumption, a compression technique is proposed based on curvelet transform, with special technique of coefficient elimination by which a higher compression ratio can be obtained without much loss in image information. Even though the coefficients neglected are large, the higher PSNR values show that curvelet has better reconstruction performance.
2. Performance evaluation metrics The basic objective of image compression is the reduction of size for transmission or storage while maintaining suitable quality of reconstructed images. Good compression schemes having a lower MSE and high PSNR. With the application of an encryption algorithm to an image, its pixels values change when compared with the original image. A good encryption algorithm must make these changes in an irregular manner and also maximize the difference in pixels values between the original and the encrypted images. Also, to get a good encrypted image, it must be composed of totally random patterns that do not reveal any of the features of the original image. The encrypted image has to be independent of the original image. It should have a low correlation with the original image [21,2,10,16]. 2.1. The image compression evaluation metrics The reconstruction quality of the compressed image can be measured in PSNR and MSE in dB. 2
PSNR ¼ 10log10 where
255 MSE
ð1Þ
PW PH MSE ¼
i1
f ji ðxij xij Þ W H
ð2Þ
where xij and f xij denotes the original and reconstructed pixel, respectively, and the images are of size W H. A lower value for MSE means lesser error, and as seen from the inverse relation between the MSE and PSNR, this translates to a high value of PSNR. Logically, a higher value of PSNR is good because it means that the ratio of Signal to Noise is higher. Here, the ‘signal’ is the original image, and the ‘noise’ is the error in reconstruction. So, a compression scheme having a lower MSE and higher PSNR can be recognized as a better one. 2.2. The image encryption evaluation metrics In this section, we evaluate the ability of the encryption algorithm to substitute the original image with uncorrelated encrypted image. Theoretical analyses for the secured image encryption on the basis of key space analysis, statistical analysis, histogram analysis, information entropy analysis, correlation analysis and differential analysis confirm that to minimize the possibility of brute force attack for decryption. 2.2.1. Key space analysis The key space should also be suitably large to make brute-force attack not feasible. 2.2.2. Statistical analysis An ideal cipher should be strong against any statistical attack, so statistical analysis on cipher-text is of crucial importance for a cryptosystem. In order to prove the security of the proposed image encryption scheme, the following statistical tests are performed. 2.2.2.1. Histogram analysis. To prevent the access of information to attackers, it is important to ensure that encrypted and original images do not have any statistical similarities. The histogram analysis clarifies that pixel values of image [4] are distributed. The histogram of original image contains great sharp rises followed by sharp declines and the histograms of the encrypted images for different round have uniform distribution which is significantly different from original image and has no statistical similarity in appearance. Therefore, it does not provide any clue for statistical attack. 2.2.2.2. Correlation analysis. The correlation between two vertically adjacent pixels, two horizontally adjacent pixels and two diagonally adjacent pixels in plain-image and cipher image respectively [4]. Then, calculate their correlation coefficient using the following two formulas:
cov ðx; yÞ ¼ Eðx EðxÞÞðy EðyÞÞ rxy
cov ðx; yÞ ¼ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi ; DðxÞ DðyÞ
ð3Þ ð4Þ
where x and y are the values of two adjacent pixels in the image. In numerical computations, the following discrete formulas were used:
EðxÞ ¼
N 1P xi N i¼1
ð5Þ
DðxÞ ¼
N 1P ðxi EðxÞÞðy EðyÞÞ N i¼1
ð6Þ
cov ðx; yÞ ¼
N 1P ðxi EðxÞÞðyi ðEðyÞÞ N i¼1
ð7Þ
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If the correlation of the encrypted image are nearest to zero then it inform good encryption quality.
Table 1 Run test specification.
2.2.3. Key space analysis Key space size is the total number of different keys that can be used in the cryptography. Cryptosystem is totally sensitive to all secret keys. A good encryption algorithm should not only be sensitive to the cipher key, but also the key space should be large enough to make brute-force attack infeasible. 2.2.4. Differential analysis In general, a desirable characteristic of an encrypted image is being sensitive to the little changes in a plain image (e.g. modifying only one pixel). Adversary can create a small change in the input image to observe changes in the result. By this method, the meaningful relationship between original image and cipher image can be found. If one little change in the plain-image can cause a significant change in the cipher-image, with respect to diffusion and confusion, then the differential attack actually loses its efficiency and becomes almost useless. The NPCR [11] measures the percentage of the number of different pixels to the total number of pixels in these two images. UACI [11] measures the average intensity of differences between the two images. The higher the values of NPCR and UACI are, the better the encryption. The MAE between these two images is defined in
MAE ¼
WP H 1 P jcði; jÞ pði; jÞj W H j¼1i¼1
ð8Þ
Consider two cipher-images, C1 and C2, whose corresponding plainimages have only one pixel difference. The NPCR of these two images is defined in
P
i;j Dði; jÞ 100% W H
NPCR ¼
ð9Þ
where W and H are the Width and Height of the image and D(i, j) is defined as
Dði; jÞ ¼
0; if C1ði; jÞ ¼ C2ði; jÞ 1; if C1ði; jÞ – C2ði; jÞ
P c1ði; jÞ c2ði; jÞ 1 100% W H i;j 255
M1 P i¼0
pðmi Þlog
1 pðmi Þ
1 2 3 4 5 6+
2315–2685 1114–1386 527–723 240–384 103–209 103–209
test the randomness of a sample sequence with the length of 20,000 bits of the encrypted image as follows: 1. Mono bit test. A mono bit test first counts the number of ‘‘1’’ in the 20,000 bit stream. Denote this quantity by X. if 9725 < X < 10,275, then the test is passed. 2. Poker test. Poker test firstly divides the 20,000 bit-stream into 5000 consecutive 4-bit segments. Count and store the number of occurrences of the 16 possible 4-bit values. Denote f(i) as the number of each 4-bit value i, where 0 6 i 6 15. Evaluate the following value:
X¼
n 16 P ½f ðiÞ2 5000 5000 i¼1
ð13Þ
The test is passed if 2.16 < X < 46.17. 3. Runs test. A run is defined as a maximal sequence of consecutive bits of either all ‘‘1’’or all ‘‘0,’’ which is part of the 20,000 bitstream. The incidences of runs of all lengths in the bit-stream should be counted and stored. The test is passed if the runs that occurred are within the corresponding interval specified in Table 1. Note that for the purpose of this test, runs of length greater than 6 are considered to be of length 6 in Table 1 in [26] 4. The Long run test. Find the longest run in the 20,000 bits. If the length of the longest run in the bit-stream of 20,000 bit (both of one and zero) is smaller than 26, the test is passed. 3. Literature review 3.1. Chaos-based image encryption schemes
ð11Þ
2.2.5. Information entropy analysis It is well known that the entropy H (m) of a message source m can be measured by
HðmÞ ¼
Required interval
ð10Þ
Another measure, UACI, is defined by the following formula:
UACI ¼
Length of run
ð12Þ
where M is the total number of symbols mi 2 m; p(mi) represents the probability of occurrence of symbol mi and log denotes the base 2 logarithm so that the entropy is expressed in bits. For a random source emitting 256 symbols, its entropy is H(m) = 8 bits. for the different cipher-image, the corresponding entropies should be nearest 7.8 to 8.0. This means that the cipher-images are close to a random source and the proposed algorithm is secure against the entropy attack. 2.2.6. FIPS 140 testing FIPS 140-2 randomness tests are four types: Mono-bit, Poker, Runs tests and Long Run tests. Each of the tests was designed to
Recently, a widely studied example of image encryption is based on chaos theory which is well established, simple but complicated dynamics. The chaos functions are used to describe the nonlinear dynamical systems. Chaos function have several interesting properties, these function are very sensitive to the initial conditions which make its importance in data security method. These functions generate random iterative values, these random iterative values are limited between bounds convergence of the iterative values after any value of iteration can never be seen. In [14], symmetric encryption scheme based on 2D chaotic map is proposed. A two or higher dimensional discretized chaotic maps is adopted for pixel permutation together with 1D map for diffusion. The encryption method called CKBA (Chaotic Key Based Algorithm) was proposed in [13]. The algorithm first generates a time series based on a chaotic map, and then uses it to create a binary sequence as a key. This method is very simple but has obvious defects in security, as pointed out lately in [28,29], this method is very weak to the chosen/known-plain-text attack using only one plain-image, and moreover its security to brute-force attack is also questionable. Color image encryption based on one-time keys and robust chaotic maps was proposed in [19]. One time key cryptosystem based on two robust chaotic maps is designed. The proposed cryptosystem has higher security due to an extremely large key space
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and proposed algorithm combines good confusion and diffusion properties. The analysis of nonlinear chaotic algorithm (NCA) map and a no. of attacks were proposed in [9]. The weaknesses of image encryption algorithm based on chaotic map describe on the basis of statistical and plain text attacks. Image encryption scheme based on 3D baker with dynamical compound chaotic sequence cipher generator was proposed in [33]. Divide dynamic block of 3D baker map by using compound chaotic map, and compare with 2D baker map. The 3D baker scheme is 2–3 time faster of 2D baker map [33]. The proposed scheme used in real-time secured image transmission. The cryptanalysis of an image encryption scheme based on a compound chaotic sequence was proposed in [3]. This work combines the confusion/diffusion in single unit for image encryption. The two-dimensional map is proposed in [5]. It is a process of stretch-and-fold. Firstly a square image is divide two parts according diagonal. Secondly each part of image is stretch respectively and joins a line. Lastly the line is fold over to a new square image of same size. The results of simulation show that it can be used in real-time image encryption applications. The efficient chaos based feedback stream cipher for image cryptosystems was proposed in [12]. The proposed stream cipher is based on the use of a chaotic logistic map and an external secret key of 256-bit. The initial conditions for the chaotic logistic map are derived using the external secret key by providing weightage to its bits corresponding to their position in the key. Several test images are used for inspecting the validity of the proposed ECBFSC. According to [20], firstly, chaotic 3D permutation is meaningless if homogenous plain images with identical pixel values are encrypted. In this case, security of the scheme relies merely on a simple diffusion process. Moreover, if a pixel value in the plain images is 0, then the underlying diffusion operation is also useless. As a result, a key recovery attacks is proposed in such a way that recovers the initial condition of logistic maps according to the gray code. Apparently, the encryption of homogenous plain image is an arbitrarily insufficient issue. However, in [7], this leads to the problem that the scheme is eventually broken with chosen plaintext attacks discussed. Guan et al. employed the 2D chaotic cat map [36] while Lian et al. employed the 2D standard map [27] for their cryptographic implementation. Fridrich’s framework adopts 2D permutation together with simple diffusion process. In 2004, some of mostly used 2D chaotic maps have also been spatially extended to higher dimensional version such as 3D cat map [7], baker map [34] and standard map [25]. In [17], proposes a new image encryption algorithm using a large pseudorandom permutation which is combinatorial generated from small permutation matrices based on chaotic maps. The random-like nature of chaos is effectively spread into encrypted images by using the permutation matrix. The proposed encryption scheme provides comparable security with that of the conventional image encryption schemes based on baker map or logistic map. In [30], perform of a quantitative cryptanalysis on the performance of ciphers against plaintext attacks. 3.2. Embedding image compression with encryption The basic idea for encryption of digital images is to selectively encrypt some important parameters/data determining the compression stage. According to the compression scheme based, some typical image encryption schemes can be classified as follows. In [1], the possibility of encrypting only the quad-tree decomposition structure of wavelet – packet based compression was discussed.
In [24], methods for encrypting medical image data selectively are discussed in special domain and frequency domain respectively. The methods proposed in [24] were analyzed and improved in [8]. In [18], an efficient joint compression and selective encryption scheme was proposed. The compression scheme is based on Set Partitioning in Hierarchical Trees (SPIHTs) of wavelet coefficients. Confidentiality of the image data is achieved by encrypting only the significance bits of individual’s coefficients. In [15], an efficient selective encryption for JPEG 2000 images was proposed. The schemes uses a secret key and a mapping function to generate a private initial table to encrypt the selected DWT code blocks in the entropy coding stage of JPEG 2000. A novel scheme for secure Internet image transmission was proposed in [22]. The feature of the proposed scheme is joint application between image compression and image encryption. For source coding, we implement Discrete Wavelet Transform (DWT) and for channel coding, we utilize block cipher Data Encryption Standard (DES) algorithm. Simulation results show that the proposed method significantly enhances security for image transmission over Internet as well as improves the transmission rate but not the reconstruction quality of the image. 3.3. Discrete logarithms based encryption schemes In their seminal paper [32], Diffie and Hellman introduced the notion of public key cryptography. They described how two entities can agree on a common secret key by communicating over an insecure channel. This is known as the Diffie–Hellman key agreement protocol. The security of the protocol is related to the apparent difficulty of computing discrete logarithms modulo of a large prime number p, i.e. given two numbers (g mod p) and (gxmod p), it seems to be infeasible to compute x for general large enough p. A few years later, Rivest et al. [23] proposed a public key encryption scheme and a digital signature scheme, the security of which is related to factoring a large integer. The papers [32,23] laid the foundations of public key cryptography. Since their appearance, many other schemes have been proposed that are based on the Integer Factorisation Problem (IFP) and the Discrete Logarithm Problem (DLP), such as the El-Gamal encryption and signature scheme [31] and the Digital Signature Algorithm (DSA) [6]. The development of newer algorithms for session key exchange by public key cryptography is the fear that the older and more established schemes might suddenly be broken due to some new algorithm/method, which is discovered to attack the underlying hard problem. As such, research has been underway to find public key algorithms based on other hard problems besides the Integer Factorization Problem (IFP) and the Discrete Logarithm Problem (DLP).
Table 2 RSA so far solved and yet to be solved. Key sizes (bits)
Data achieved
MIPS year
Algorithm
332 365 398 428 431 465 512 1024
April 1991 April 1992 June 1993 April 1994 April 1996 February 1999 August 1999 To be solved
7 75 83 500 1000 2000 8000 To be solved
Quadratic sieve Quadratic sieve Quadratic sieve Quadratic sieve GNFS GNFS GNFS To be solved
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K. Gupta, S. Silakari / Advances in Engineering Software 49 (2012) 29–42 Table 3 Key size ratio for ECC/RSA. Symmetric key size
ECC key size
RSA/DH key size
Key-size ratio
80 128 192 256
163 256 384 512
1024 3072 7680 15,360
1:6 1:12 1:20 1:30
Permutation by 3D Standard Map
Diffusion Template Standard
Final Diffusion Template
Fig. 1. Diffusion template.
1. RSA is based on Integer Factorization algorithm and hence factoring performance can be used as a benchmark against which the security of RSA can be evaluated. As shown in Table 2, the latest challenge in the RSA with a key length of 1024 bits. A striking feature about the Table 2 is the algorithm used to break RSA. 2. Attacks on RSA can be overcome by increasing the key size, but this makes the key generation process more complex and time consuming. This also increases the time of the encryption and decryption algorithms, thereby increasing the storage requirement. 3. The primary reason for selecting ECC for secured session key exchange for solving ECDLP mathematical problem takes fully exponential time. In contrast, the RSA based on take subexponential time to solve Integer factorization. 4. The following Table 3 gives the key sizes recommended by the National Institute of Standards and Technology to protect keys used in conventional encryption algorithms like the (DES) and (AES) together with the key sizes for RSA, Diffie– Hellman and Elliptic Curves that are needed to provide equivalent security. 5. The major advantage of ECC compared to other public key schemes based on either the IFP or the DLP is that ECC based cryptosystems offer equivalent security to those older cryptosystems at much shorter key lengths. This results in faster operations, lower processing requirements and even low space and bandwidth conservation. So, the ECC are obvious choice for session key exchange in secured manner. Although many image encryption schemes have been proposed, some of them are too weak to resist various cryptanalytic attacks. Many schemes using compression with encryption have some problem related to security as well as reconstruction of image quality. The ECDLP is more suitable for session key exchange because it is faster compared to integer factorization problem. In our study, there is no comprehensive model found that provides security as well as fast, cost effectiveness. In the whole, the cryptanalytic work on the proposed schemes is not enough. Much more work is needed to be done to design more secure, fast and efficient image transmission schemes.
4.1. Diffusion template According to the proposal the diffusion template must have the same size as main image. Let the main image have m number of rows n number of columns then the diffusion template is created as follows:
ði; j; kÞ ¼ round
255 j n
where 1 6 i 6 m, 1 6 j 6 n and 1 6 k 6 3. Eq. (14) form the matrix with all rows filled with linearly spaced number in between 0 to 255. The sequence is randomized by 3D standard map in discrete form. The 3D standard map randomizes the pixels by reallocating it in new position by utilizing its property of one to one mapping. Fig. 1 show the final diffusion template by using 3D standard map. 0
i ¼ ði þ jÞmod m n 0 mod n j ¼ j þ k þ K1 sin i 2 pi p p 0 þ K2 sin j mod p k ¼ k þ K1 sin i 2 pi 2 pi
0
0
ð17Þ
0
4.2. 148-bit session key generation process The proposed method has a large number of variables which can be used as key parameters but to avoid the exceptionally large key and decreased key sensitivity, the parameter which does not having great affects on encryption are avoid or scaled. The selected key parameters and their length are given below Step 1. Diffusion template shuffling
Step 2. Diffusion template offset value
Step 3. Diffusion template variables
Dk1Dk2 ¼ 8 þ 8 ¼ 16 bits: 1. The image is encrypted by the chaos based encryption using cascading of 3D cat map and standard map. 2. The chaos based encryption comes under the symmetric cryptography which depends on session key, so we are using Elliptic Curve Cryptography method for session key exchange in secure manner. 3. Before encryption, the image is compressed by using curvelet transform for eliminate the redundancy from the color image as well as better utilization of encryption and fast transmission.
ð16Þ
I0diff ði ; j ; k Þ ¼ Idiff ði; j; kÞ
DxDyDz ¼ 8 þ 8 þ 2 ¼ 18 bits: The proposed cryptosystem divide into three parts as shown in Fig. 2.
ð15Þ
where the K1, K2 are the integers, p = 3 for the case of color image and i0 , j0 k0 shows the transformed location of i, j, k
Ds ¼ 8 bits: 4. Proposed Compressed Hybrid Image Cryptosystem
ð14Þ
Step 4. Sliced RGB plane Shuffling
Ss ¼ 8 bits: Step 5. Sliced RGB plane offset values
SxSy ¼ 8 bits: Step 6. Sliced RGB Plane Variables
SpSq ¼ 8 þ 8 ¼ 16 bits:
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Read ith Row of Red
Rotate Red Plane Vertically
Compressed N×N Image
Rotate Green Plane Horizontally
Rotated Image
Read ith Row of Green
Remove Coefficients below Cutoff C1 = C > cutoff
Create a Plane by These 3 rows
Generate Diffusion Template Standard
Read ith Row of Blue
Blue Plane remain same
Combine the Planes to form shuffled image
Set cutoff Threshold
Repeat the Process for m Times
Permutation by 3D Standard Map Shuffle the Plane by 2D cat map
by N = CPR*image size
Final Diffusion Template
Cutoff = Curvelet coefficient (N)
Find the new Arrange in descendant orde
Shuffled Image
location of pixel I (i, j, k) by 3D cat map let (i’ j’, k’)
Find the new location of pixel (i’, j’, k’) by 3D Standard map
Calculate the Curvlet Coefficient Repeat The Process for all pixels Input Plain image M×N
Repeat the Process for n Times
Perform EXOR with Final Diffusion Template
Session Key ha Exchange ing E Using ECC
Reconstruct Plain image M×N
Perform inverse Curvelet
Compressed image
Open Channel N/W
Perform inverse Decryption Algorithm
Fig. 2. Compressed Hybrid Cryptosystem.
Step 7. Final Confusion shuffling
Cs ¼ 8 bits:
Step 8. Confusion offset of cat map
CxCyCz ¼ 8 þ 8 þ 2 ¼ 18 bits:
Final Encrypted Image
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Step 9. Confusion cat map variables
35
Step 4: Generate Final confusion stage by two cascade 3D maps first by cat map then by standard map. So the transformation of location (i, j, k) into (i00 , j00 , k00 ) is performed by following equations.
CaCb ¼ 8 þ 8 ¼ 16 bits: Step 10. Confusion offset of standard map
0 i ¼ ð1 þ ax az by Þ i þ az j þ ðay þ ax az þ ax ay az by Þ k mod m
0 j ¼ ðbz þ ax by þ ax az by bz Þ i þ ðaz bz þ 1Þ j þðay az þ ax ay az by bz þ ax az by þ ax ay by þ ax Þ k mod n
0 k ¼ ðax bx by þ by Þ i þ bx j þ ðax ay bx by þ ax bx þ ay by þ 1Þ k mod p
Cx0 Cy0 Cz0 ¼ 8 þ 8 þ 2 ¼ 18 bits: Step 11. Confusion standard map variables
Ck1Ck2 ¼ 8 þ 8 ¼ 16 bits: Final key structure
DsDxDyDzDk1Dk2SsSxSySpSqCs
00
0
0
i ¼ ½ði þ k Þmod m
0
CxCyCzCaCbCx Cy0 Cz0 Ck1Ck2 Total bits ¼ 8 þ 18 þ 16 þ 8 þ 8 þ 16 þ 8 þ 16 þ 16 þ 18 þ 16 ¼ 148 bits: 4.3. Novel algorithm for color image encryption/decryption based on 3D cat and standard maps In this algorithm, we generate diffusion template using 3D standard map and rotate image by using vertically and horizontally red and green plane of the input image. We then shuffle the red, green, and blue plane by using 3D cat map and 3D standard map. Finally the image is encrypted by performing XOR operation on the shuffled image and diffusion template.
h n i 00 0 0 00 mod n j ¼ i þ j þ K1 sin i 2p h p p i 00 0 00 00 mod p k ¼ k þ K1 sinði Þ þ K2 sin j 2p 2p 00
00
00
ði ;j ;k Þ ¼ I0new ði;j;kÞ
where ax, ay, az, bx, by, bz and K1, K2 are integers >0 Step 5: Each confusion step is followed by diffusion obtained by XOR operations performed between each pixels of Iconf and diffusion Idiff.
Iencp ¼ Iconf Idiff
ð20Þ
4.3.2. Algorithm for color image decryption 4.3.1. Algorithm for color image encryption Step 1: The main image is divided into three separate images IR, IG and IB as follows:
IR ðx; yÞ ¼ Iðx; y; 1Þ IG ðx; yÞ ¼ Iðx; y; 2Þ
0
IB ðx; yÞ ¼ Iðx; y; 3Þ where 1 6 x 6 m and 1 6 y 6 n Step 2: The red and green image are transform vertically and horizontally respectively. The blue image remains same and reconstructs the new image.
Inew ðx; y; 1Þ ¼ I0R ðx; yÞ
Inew ðx; y; 3Þ ¼ I0B ðx; yÞ m mod m; y I0R ðx; yÞ ¼ IR x þ 2 n I0G ðx; yÞ ¼ IR x; y þ mod n 2
Iretransf ði;j;kÞ¼Iencp ði;j;kÞ
Step 3: Perform the first level confusion by using 2D cat map. Slice the plane normal to R, G, and B planes by
k ¼ ðq j þ r y þ ðp q þ 1Þ kÞmod n
I0dencp ¼ Iretrnsf Idiff
ð21Þ
I0SRGB ðj; kÞ ¼ I0dencp ði; j; kÞ
where 1 6 i 6 m, 1 6 j 6 n and 1 6 k 6 3. 0
where ax, ay, az, bx, by, bz and K1, K2 are integers >0. Each confusion step is followed by diffusion obtained by XOR operations performed between each pixels of Iretrnsf and diffusion Idiff.
Step 3: Performing inverse of first level confusion slicing the plane normal to R, G, B planes
0
I0new ði; j; kÞ ¼ I0SRGB ðj ; k Þ ¼ Inew ði; j; kÞ
j ¼ ðj þ rx þ ry þ p kÞmod m
00
0 0 0 k¼ ðax bx by þby Þi þbx j þðax ay bx by þax bx þay by þ1Þk mod p
I0B ðx; yÞ ¼ IB ðx; yÞ
0
00
i ¼ ½ði þk Þmod m n 0 00 00 0 j ¼ i þj þK1sin i mod n 2pi p p 0 00 0 0 þK2sin j mod p k ¼ k þK1sin i 2pi 2pi
0 0 0 i ¼ ð1þax az by Þi þaz j þðay þax az þax ay az by Þk mod m
0 0 j ¼ ðbz þ ax by þ ax az by bz Þ i þ ðaz bz þ 1Þ j þ ay az þ ax ay az by bz þ ax az by þ ax 0 ay by þ ax Þ k mod n
Inew ðx; y; 2Þ ¼ I0G ðx; yÞ
0
Step 1: Generate the diffusion template in same way as in encryption section. Step 2: Re-transformation of location is done by two cascaded 3D maps firstly by standard map then by cat map. So the retransformation of location (i00 , j00 , k00 ) into (i, j, k) is performed by following equations
ð18Þ ð19Þ
where j0 and k0 are obtained by 2D cat map and p and q are integer >0 and rx,ry are offset integer such that 0 6 rx 6 m, 0 6 ry 6 n.
for each value of i, j changed from 0 to m, k changed from 0 to 3 De-shuffling the sliced plane
IDRGB ðj; kÞ ¼ I0SRGB ðj; kÞ where j0 and k0 are obtained by 2D Cat map given below
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K. Gupta, S. Silakari / Advances in Engineering Software 49 (2012) 29–42 0
j ¼ ðj þ r x þ r y þ p kÞmod m 0
k ¼ ðq j þ ry þ ðp q þ 1Þ kÞmod n
ð22Þ ð23Þ
where p and q are integers >0, and rx, ry are offset integers such that 0 6 rx 6 m and 0 6 ry 6 n Recombining the planes for forming 3D matrix for next operation
I0 ði; j; kÞ ¼ I0DRGB ðj; kÞ Step 4: Re-rotating the image planes Dividing main image into three separate images IR, IG and IB as follows
IG ðx; yÞ ¼ I0 ðx; y; 2Þ IB ðx; yÞ ¼ I0 ðx; y; 3Þ where 1 6 x 6 m and 1 6 y 6 n. Scrolling the red plane vertically
m IR ðx; yÞ ¼ IR x þ mod m; y 2 n IG ðx; yÞ ¼ IR x; y þ mod n 2 Blue plane remain intact.
IB ðx; yÞ ¼ IB ðx; yÞ Step 5: Next recombination of planes are performed to form final decrypted image
Ifinal ðx; y; 1Þ ¼ IR ðx; yÞ Ifinal ðx; y; 2Þ ¼ IG ðx; yÞ
4.4. Algorithm for image compression based on curvelet transforms This research proposes a novel image compression algorithm using curvelet transform. The original image was decomposed into curvelet coefficients using fast discrete curvelet transform, after that the different scales of quantized curvelet coefficients were selected for lossy compression and arranged in descending order. Then we set the cutoff threshold for curvelet coefficients. The proposed method was compared with image compression method based on wavelet transform. Experimental results show that the compression performance of our method gains much improvement based on PSNR and MSE. Moreover, the algorithm works fairly well for declining block effect at higher compression ratios. Step 1: Calculate the cuvelet coefficient of the image planes using following equations
f ðxÞwj;l;k ðxÞdx
ð24Þ
where R denotes the real line. Step 2: Calculate the size of compressed image according to given compression ratio (CPR). Step 3: Arrange the curvelet coefficients in descending order C. Step 4: Find out the cutoff threshold for curvelet coefficients (CL) as given below
N ¼ CPR Image size Cutoff ¼ CL ðNÞ where CL is the curvelet coefficients array arrange in descending order. Step 5: : Remove all the coefficients below cutoff
C1 ¼ C > Cutoff
y2 ðmod pÞ ¼ ðx3 þ ax þ bÞmod p
ð25Þ
P B ¼ nB G Step 4: PA, p, a, b and generator point G is made public. Step 5: The cipher text is generate by
P c ¼ ½ðkGÞ; ðPM þ kP B Þ ¼ fðx1 ; y1 Þ; ðx2 ; y2 Þg
Ifinal ðx; y; 3Þ ¼ IB ðx; yÞ
R2
The symmetric key cryptography totally depends on the session key (private key). ECC is well known method for public key cryptosystem as it is having the highest strength-per-bit compared to other public key method; this algorithm addresses the session key encryption using ECC.
where a, b are two integers which satisfy 4a3 + 27b2 – 0(mod p). Then the elliptic group, Ep(a, b), is the set of pairs (x, y), where 0 6 x, y < p, satisfy the Eq. (12). The smallest value of n for which n ⁄ G = O is a very large prime number. (⁄Here nG is special multiplication called Multiplication over an elliptic curve group). Step 3: User A selects a private key, nA < n and compute the public key PA as:PA = nA ⁄ G Similarly for user B public key is
Scrolling the green plane horizontally
Z
4.5. Algorithm for 148-bit session key encryption using ECC
Step 1: The 148-bit session key generated by procedure 4.2, this is assume plain message PM. Step 2: We consider an Elliptic Curve over a finite field associated with a prime number p > 3 whose equation is
IR ðx; yÞ ¼ I0 ðx; y; 1Þ
Cðj; l; kÞ ¼
Step 6: : Perform inverse curvelet transform of C1 to get compressed image.
where k is random positive integer Step 6: On receiving of cipher text user a finds the original points using the equation
P M ¼ ðPM þ kPB Þ nB ðkGÞ ¼ ðPM þ kðnB GÞÞ nB ðkGÞ ¼ PM Here ðPM þ nB PÞ and (kG) are directly taken from Pc. 5. Result and performance analysis Hence the proposed model constitutes compression, encryption and secured session key exchange along with the transmission of image. Performance was measured on a machine with Intel core 2 Duo 2.00 GHz CPU with 2 GB of RAM running on Windows XP.
Image Encoded Time T E ¼ T Comp þ T Encry þ T SkeyEncry Image Decoded Time T E ¼ T SkeyDecry þ T Decry þ T Decomp where TComp = time transform.
for
image
compression
using
curvelet
T Encry ¼ time for image encryption using chaotic map T SkeyEncry ¼ time for 148-bits session key encryption using ECC T SkeyDecry ¼ time for 148-bits session key decryption using ECC T Decry ¼ time for image decryption using chaotic map T Decomp ¼ time for image decompression using curvelet transform The model has been rigorously examined over the prevalent standard test has encouragingly succeeded to pass most of them like key sensitivity analysis, key space analysis, statistical analysis, differential analysis, entropy analysis, FIPS-140 test, PSNR analysis as shown in Table 4 and Figs. 3–5, for fast, cost effective and secured image transmission, which was the key problem statement for this work.
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K. Gupta, S. Silakari / Advances in Engineering Software 49 (2012) 29–42 Table 4 Result for Compressed Hybrid Cryptosystem. R=1
R=2
R=3
R=4
R=8
R = 10
R = 16
R = 32
Lena 256 256 for compression ratio = 1:20 using curvelet transform PSNR MSE NPCR UACI MAE Encoded time Decoded time Entropy Run 0 Run 1 Poker test Mono-bit test
40.41 7.71 99.58 17.54 96.86 2.8332 2.3806 7.9993 17 14 342 9879
40.41 7.71 99.53 17.71 96.39 2.8929 3.5806 7.9946 17 15 39 9871
40.41 7.71 99.59 17.71 96.86 2.9317 3.105 7.999 15 15 30 9964
40.41 7.71 99.63 17.56 96.97 3.0517 2.7368 7.999 14 18 13 10036
40.41 7.71 99.59 17.62 96.76 3.9572 4.2244 7.9991 16 14 13 10008
40.41 7.71 99.62 17.65 96.65 5.0517 4.3425 7.999 15 13 15 9964
40.41 7.71 99.59 17.54 96.57 8.150 6.2208 7.9992 12 14 15 9942
40.41 7.71 99.59 17.51 96.66 10.513 10.378 7.9991 14 16 13 9941
Lena 512 512 for compression ratio = 1:20 using curvelet transform PSNR MSE NPCR UACI MAE Encoded time Decoded time Entropy Run 0 Run 1 Poker test Mono-bit test
47.24 0.883 99.60 17.51 96.88 6.202 4.624 7.993 17 16 343 9868
47.24 0.883 99.59 17.70 97.10 6.417 6.323 7.994 13 12 33 10003
47.24 0.883 99.59 17.59 97.25 7.196 6.629 7.999 13 14 27 10038
47.24 0.883 99.62 17.56 97.31 7.701 5.637 7.999 18 14 11 9941
47.24 0.883 99.62 17.47 97.22 8.778 7.795 7.999 14 14 13 9976
47.24 0.883 99.59 17.50 97.21 9.794 8.512 7.9991 15 14 10 9989
47.24 0.883 99.60 17.44 97.19 11.237 8.729 7.9992 15 16 14 9999
47.24 0.883 99.58 17.54 97.21 16.017 13.703 7.9991 17 14 19 9897
Lena 1024 1024 for compression ratio = 1:20 using curvelet transform PSNR MSE NPCR UACI MAE Encoded time Decoded time Entropy Run 0 Run 1 Poker test Mono-bit test
55.18 7.049 99.618 17.307 97.373 25.594 16.453 7.999 17 16 366.16 9824
55.18 7.049 99.562 17.535 97.261 25.942 20.475 7.994 13 12 38.55 9963
55.18 7.049 99.593 17.417 97.737 26.848 20.657 7.999 15 14 25.34 9972
55.18 7.049 99.625 17.387 97.70 27.125 21.402 7.999 14 15 8.8 9909
55.18 7.049 99.615 17.348 97.536 28.091 21.794 7.9990 16 14 10.04 10120
55.18 7.049 99.606 17.414 97.608 30.069 22.241 7.9991 15 16 6.745 9955
55.18 7.049 99.602 17.245 97.423 30.541 23.025 7.9993 15 16 12.684 10045
55.18 7.049 99.590 17.403 97.599 36.457 29.371 7.9991 14 13 13.491 9840
The test has been performed on different sizes of Lena, Baboon images from SIPI database. The image encoded and decoded times through curvelet transform are considerable for real-time application. Fig. 3 denote the encoded and decoded time of 1024 1024, 24-bit Lena color image. The poker test has passed for all rounds except round one that is shown in Table 4. Fig. 4 depicted the mono-bit test, which is passed for all round.
Fig. 5 defined the NPCR and UACI for Compressed Hybrid Cryptosystem i.e. acceptable encryption performance for real time application. The Table 5, compare the result of Wang et al. [35] and Lien at all [20] work based on chaotic map and results obtained from NPCR shows that the sensitivity of encryption scheme for little changes in the input image is under 0.01%. Fig. 6 comparison of NPCR with Wang et al. [35] and Lian et al. [20]. The results demonstrate that a
Fig. 3. Image encoded and decoded time through curvelet transform.
Fig. 4. FIPS-140 Poker test.
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Fig. 5. NPCR and UACI for Compress Hybrid Cryptosystem.
swift change in the original image will result in a negligible change in the ciphered image. 5.1. Key space analysis The strong point of the proposed algorithm is the generation of the permutation sequence with the chaos sequence. The key space should also be suitably large to make brute-force attack not feasible. In the proposed algorithm, we use 148 bit key (37 Hex numbers). It is observed in Figs. 7 and 8 that with slight variation in
Fig. 6. Comparison of NPCR with Wang et al. [35] and Lien et al. [20].
Table 5 Comparison of NPCR with Wang et al. [35] and Lien et al. [20]. Name of image
No. of rounds
Proposed
Lien et al.
Wong
Lena 256 256 (24-Bit color)
1
0.9955
0.0003
0.9944
2 3 4 5 6
0.9957 0.9959 0.9961 0.9961 0.9962
0.1776 0.5662 0.9873 0.9959 0.9961
0.9961 0.9961 0.9961 0.9961 0.9962
Fig. 7. Input image encrypted with 0304002030402 301011010110D2833020202 and decrypted with 03040020304020301011010110D2833020203.
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Fig. 8. Input Lena image encrypted 03040020304020301011010110D2833040404 and decrypted by 03040020304020301011010110D2833040405.
Fig. 9. Histogram for red, green and blue plane of original and encrypted image for R = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 10. Histogram for red, green and blue plane of encrypted image for R = 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. Histogram for red, green and blue plane of encrypted image for R = 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 12. Correlation for horizontal, vertical and diagonal for 256 256 Lena image of round R = 1.
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Fig. 13. Correlation for horizontal, vertical and diagonal for 256 256 Lena image of round R = 2.
the initial condition of the chaotic sequence, the resultant image after decryption differs a lot from the original image. So it is very difficult to breach the security by brute force attack. 5.2. Histogram analysis To prevent the access of information to attackers, it is important to ensure that encrypted and original images do not have any statistical similarities. The histogram analysis clarifies that pixel values of image are distributed. A number of images are encrypted by the encryption schemes under study and visual test is performed. As shown in Figs. 9–11. The histogram of original image contains great sharp rises followed by sharp declines and the histograms of the encrypted images for different round have uniform
Table 6 Encrypted coordinate of 4567D020202 with ECC. (8, 20) (24, 20) (24, 20) (14, 5) (23, 27) (2, 10) (6, 2)
(15, 4) (2, 19) (5, 4) (19, 26) (22, 26) (22, 3)
the
session
(2, 19) (8, 15) (2, 20) (2, 19) (23, 13) (7, 19)
key
97056123456837619072A4BE3C
(14, 24) (20, 14) (2, 15) (23, 13) (19, 2) (23, 26)
(5, 4) (8, 5) (14, 15) (6, 22) (26, 6) (6, 2)
(2, 16) (16, 4) (14, 15) (7, 19) (10, 7) (6, 2)
Fig. 14. Encrypted coordinate of the session key 97056123456837619072 A4BE3C4567D020202 with ECC.
distribution which is significantly different from original image and has no statistical similarity in appearance. So, the proposed algorithms do not provide any clue for statistical attack. The encrypted image histogram, approximated by a uniform distribution, is quite different from plain-image histogram. 5.3. Correlation analysis In addition to the histogram analysis, we have also analyzed the correlation between two vertically adjacent pixels, two horizontally adjacent pixels and two diagonally adjacent pixels in plainimage/cipher-image respectively. The correlation coefficient between original and cipher image of horizontal, vertically and diagonally is calculated in Figs. 12, 13 for round = 1, 2. The 148-bits session key 97056123456837619072A4BE3C 4567D020202 encrypted with ECC parameter of P = 29, a = 4, b = 20, Gnum = 2, na = 12, nb = 13. Encrypted coordinate as shown in Table 6 and Fig. 14, show our result that achieves very large repetition cycle without reducing its security strength.
6. Conclusion In this chapter, we proposed ciphered model constitutes compression, encryption and secured session key exchange along with the transmission of image. The model has been rigorously examined over the prevalent standard test and has encouragingly succeeded to pass most of them, for fast, cost effective and secured image transmission, which was the key problem statement for this work. Hence this proposed model can be considered as a big breakthrough in the multiple application domain i.e. Military Image Database & Message Communication, Medical Imaging System & Telemedicine, Confidential Video Conferencing, Online Image Identification and Authentication, Electronic Surveillance System, Document Imaging, Image ‘CAPTCHA’, Image Registration, Geographic Information System, Distributed Secured Database, Biometric Identification and may also help in streamlining the future research in this domain. The proposed model can be used in above mentioned applications, by which the content would have considerable degree of added security and speed which in turn would prevent the data from eavesdropping as well as better utilization of transmission channel respectively.
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Kamlesh Gupta Assistant professor in Department of Computer Science & Engineering at Jaypee University of Engineering and Technology, Guna, M.P., India. He has published 14 papers in International and National journals and conferences. His research interest includes cryptography and image processing. He is pursuing Ph.D. from RGPV, Bhopal, India. He is a life member of IETE.
Dr. Sanjay Silakari Professor and Head, Department of Computer Science & Engineering at Rajiv Gandhi Technological University, Bhopal, India. He has awarded Ph.D. degree in Computer Science & Engineering. He posses more than 18 years of experience in teaching under graduate and post graduate classes. He is guiding nine Ph.D students. He has published 75 papers in referred journals and conference proceedings. He is a member of IACSIT Singapore.
