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Mar 29, 2011 - The continuation f1 is a coarse version of the original image f0(a polygonal ... tional to the number N of data, which is not invalid in the signal (O(N)) what ..... of coding of size 64 bits (8 bytes) for the case of worms a key of size ...
J. of Commun. & Comput. Eng. ISSN 2090-6234 Volume 1, Issue 1-2, 2011, Pages 1: 11

c Modern Science Publishers Copyright ⃝ www.m-sciences.com

Hybrid Methods of Image Compression-Encryption M. Benabdellah · F. Regragui · E. H. Bouyakhf.

Received: 29 March 2011 / Accepted: 27 May 2011

1 Introduction Abstract The networks have been developing strongly these last years and became inevitable for the modern The only work to date, by Johnson et al., [17] shows communication. The images transmitted on these netthat through the use of coding with side information works are particular data because of their large quanprinciples, the compression of encrypted data can thetity of information. The transmission of the images thus oretically be accomplished without loss of coding effiraises a significant number of problems which the maciency or perfect secrecy in certain scenarios. For exjority of them have no solution yet. We enumerate for ample, if the original source is Gaussian, the same efexample safety, confidentiality, integrity and the auficiency can be achieved for loss compression of an enthenticity of these data during their transmission. Some crypted source, as when encryption is effected after medical applications require a strict association between compression. In Johnson, the decompression and dethe image and its contextual data. The protection of cryption operations were performed jointly at the rehigh resolution information (high frequencies of the imceiver. However, most commercial image compression ages, details, scalable visualization) currently has a high standards accomplish compression/ decompression withdemand. During this work, our research led to the creout any supposition about encryption/decryption at ation of two new methods to make safe the transfer later stages. A variety of encryption algorithms can of images. The two methods are based on hybrid codbe employed to remove intelligibility from an uncomings: use of compression by Faber-Schauder Multi-Scale pressed archived image. This local encryption is genTransform (FMT) and encryption by the two algorithms erally performed to preserve data security without asDES and AES. The comparison of these approaches with methods of compression-encryption, such as Quadtree- suming compression at another time. However, postencryption transmission of such locally-stored images AES, DCT-partial encryption and DCT-RSA, showed stipulates compression to improve bandwidth utilizawell its good performance. tion. Keywords Multi-scale base of Faber-Schauder · FMT · Mixed Visualization · Encryption · DES · AES · compression-encryption

M. Benabdellah Facult des Sciences Juridiques, Economiques et Sociales, Universit Mohamed I, Oujda, Maroco E-mail: med [email protected] F. Regragui and E. H. Bouyakhf FSR, University of Mohamed V, Rabat, Morocco

The best would be to be able to apply asymmetrical systems of encryption so as not to have a key to transfer. Because of the knowledge of the public key, the asymmetrical systems are very expensive in calculation, and thus a protected transfer of images cannot be envisaged [19]. The symmetrical algorithms impose the transfer of the secrete key. The traditional methods of encryption images impose the transfer of the secret key by another channel or another means of communication [1]. The encryption algorithms per blocks applied to the images present two disadvantages: on the one hand,

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when the image contains homogeneous zones, all the identical blocks remain identical after the encryption. For this, the encrypted image contains textured zones and the entropy of the image is not maximal. In addition, the techniques of encryption per blocks are not resistant to the noise. In fact, an error on a coded bit will propagate important errors on the running blocks entirely [3]. The traditional methods compression-encryption have all tendencies to carry out techniques of encryption and compression in a disjoined way; this causes a problem during the decrypting and the decompression stages, especially in the case of some application domains of real time type like the emission of images by satellites or the telemedicine where time is a paramount factor. For a protected and reduced transfer of images, the algorithms of encryption images must be able to be combined with the algorithms of compression of images. The techniques of compression seek the redundancies contained in the images in order to reduce the quantity of information [8]. On the other hand, the techniques of encryption aim to remove all the redundancies to avoid the statistical attacks, which is the famous problem.

M. Benabdellah et al.

the transformation. We notice a concentration of coefficients around the outline areas, and this is confirmed by the particular aspect of the histogram. If we encrypt only his significant coefficients we will only have a small description of the multi-scale image and, with a good conditioning, we will be able to decipher and rebuild the initial image without a big debasement. In what follows, we describe the basic multi-scale construction of Faber-Schauder and we focus on the algorithm of transformation and reverse transformation. Then, we introduce the mixed-scale visualization of the transformed images and its properties. Afterwards, we speak about the compression of images by the FMT, and we explain the DES and the AES encryption algorithms. Lastly, we finish by the general diagram of the hybrid methods of the introduced compressionencryption and the results found, after the application and comparison with the methods Quadtree-AES, DCT-RSA and the DCT-partial encryption.

2 Methods 2.1 Faber-Schauder multi-scale transformation

In many methods of compression of images in grey level, the principal idea consists in transforming them so as to concentrate the piece of information (or the energy) the image in a small number of pixels. In general, the linear transformations are preferred because they allow for an analytic study [7]. Among the most used transformations, we can quote that of the cosine which is at the base of the standard of JPEG compression [16].

The Faber-Schauder wavelet transform is a simple multiscale transformation with many interesting properties in image processing. In these properties, we advertise multi-scale edge detection, preservation of pixels ranges, elimination of the constant and the linear correlation [10].

The multi-scales transformations make it possible to take into account, at the same time, the great structures and the small details contained in an image; and from this point of view, they have similarities with the human visual system. Laplacian pyramid algorithm of BurtAdelson was the first example known, but it suffers in particular from the redundancy of the representation of data after transformation [22].

f (x) = a0 +

Mallat used the analysis of the wavelets to develop a fast algorithm of multi-scales transformation of images which has same philosophy as the diagram of the laplacian pyramid, but it is most effective [9]. In this paper, we present the Faber-Schauder Multiscales Transformation (FMT), which carries out a change of the canonical base towards that of Faber-Schauder. We use an algorithm of transformation (and reverse transformation), which is fast and exact. Then, we present a method of visualization at mixed scales which makes it possible to observe, on only one image, the effect of

Let’s look now at the origins of the wavelet transform, those are to the Fourier transform. It’s well known that any 2π periodic functions f (x) can be represented by its Fourier series in the form of: ∞ ∑

(ak cos (kx) + bk sin(kx))

1

where k = 1, 2, 3, . . .. ∫ 1 2π f (x) cos (kx) dx ak = π 1 and bk =

1 π

∫

2π

f (x) sin (kx) dx. 0

However, it appeared that not all 2π periodic functions can be represented in such a way. In 1897 Du BoisReymond constructed a 2π periodic function, which Fourier series diverged at a certain point. This has lead to a question: Is there a different orthonormal system of functions, defined on [0, 1], such that for any function,

Hybrid Methods of Image Compression-Encryption

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continuous on [0, 1] its series converges uniformly on [0, 1] ?!

2.1.1 Construction of the Faber-Schauder multi-scale base

This question led to the creation of Haar system, using which we can write a series

The multi-resolution analysis [13] of L2 (R) is composed of vector spaces (Vj ) of the (linear continuous functions ) [ ] per pieces on the intervals k2j , (k + 1)2j k∈Z such as [11]:

⟨f,h0 ⟩ h0 (x) + ⟨f,h1 ⟩ h1 (x) + . . . + ⟨f,hn ⟩ hn (x) + . . . which converges to f (x) uniformly on [0, 1]. Here (a, b) stands for scalar product of a and b, and hn stand for the elements of the Haar system which are acquired from the basic function [ ) { 1 1, x ∈ 0, [ 1 2 ), h (x) = −1, x ∈ 2 , 1 . For n ≥ 1 we take n = 2j + k where j ≥ 0 and 0 ≤ k ≤ 2j and thus define hn as follows: ( ) 1 hn =2 2 h 2j x − k . Finally, to complete the hn set and to acquire an orthonormal basis for L2 [0, 1] we define h0 (x) = 1. The approximation of f (x) by the Haar series is simply the approximation of a continuous function by step functions with values equal to the mean values of f (x) on the appropriate intervals [22]. There is a serious disadvantage in the Haar system, suppose that f (x) is not simply continuous, but also has a continuous derivative. Then the approximation by step functions is completely inappropriate. To avoid it, Faber and Schauder suggested replacing hn (x) functions by their primitives, thus defining the Schauder basis. This basis may also be defined as one constructed from the mother function ∆(x), as the Haar basis was [11]. Here we have: { ∆ (x) =

2x, 2 (1 − x) ,

[ ] x ∈ [ 0, 21 ] , x ∈ 12 , 1 ,

and as in the previous case we define ∆n (x) =∆(2j x − k) for n = 2j + k and with j ≥ 0 and 0 ≤ k ≤ 2j. Then after adding ∆0 (x) = x and a constant 1 we define a Schauder basis.∑ The function f (x) is then ∞ written as f (x) = a + bx+ 1 αn ∆n (x). If we derive this expression term by term, we can see ′ then that it turns into an expression for f (x) in the Haar basis, thus solving the disadvantage of the Haar basis mentioned [6]. In general, the base of Faber-schauder, which is not a Hilbertienne base, is obtained by taking the primitives of the functions defining the base of Haar.

.......V2 ⊂ V1 ⊂ V0 ⊂ V−1 ⊂ V−2 ........ An unconditional base of each space Vj , called a canonical base, and is given by the family of functions: [ j ( )] ϕn (x) = 2−j ϕ 2−j x − n n∈Z . Where

  1 + x if − 1 ≤ x ≤ 0, ϕ (x) = 1 − x if 0 ≤ x ≤ 1,  0 otherwise. For a 2D signal, a multi-resolution analysis of L2 (R) can be built from the tensorial products of spaces Vj : Vj = Vj × Vj , and the canonical base of Vj are given by [12]: ( ) j ϕi,j (x, y) =2−j ϕ 2−j x − k, 2−j y − 1 k,k z

with ϕ (x, y) =ϕ (x) ×ϕ (y) . The images are also sequences of numbers f 0 = (fk,l )k,l∈Z in L2 (Z 2 ) representing the values of pixels. An image can be then associated with function f of V0 given by: ∑ f (x, y) = fk,l ϕ0k,l (x, y). k,l∈Z

For the construction of the Faber-Schauder base, we suppose the family of under spaces (Wj )j∈Z of L2 (R2 ) such as Vj is the direct sum of: Vj+1 and Wj+1 : ⊕ Vj =Vj+1 Wj+1 , ⊕ ⊕ Wj+1 =Vj+1 ×Wj+1 Wj+1 ×Vj+1 Wj+1 ×Wj+1 . The space base Wj+1 is given by: ( ) j+1 j j ψl,k,l =ϕj2k+1 ×ψlj+1 , ψ2,k,l =ψkj+1 ×ϕj2l , ψ3,k,l =ψkj+1 ×ψlj+1 And the unconditional base and Faber-Schauder multiscale of L2 (R2 ) is given by: m m m (ψ1,k,l , ψ2,k,l , ψ3,k,l )k,l,m∈Z . ∑ 0 ϕ0k,l (x, y) can be A function of V0 : f (x, y) = k,l∈Z fk,l broken up in a single way according to V1 and W1 : ∑ ∑ 1 1 11 1 f (x, y) = fk,l ψk,l (x, y) + |gk,l ψk,l (x, y) k,kz

+

21 2 gk,l ψk,l

k,k,z 31 3 (x, y) +gk,l ψk,l

(x, y) |.

. k,k,z

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The continuation f 1 is a coarse version of the original image f 0 (a polygonal approximation off 0 ), while g 1 = (g 11 , g 21 , g 31 ) represents the difference in information between f 0 and f 1 . g 11 (respectively, g 21 ) represents the difference for the first (respectively the second) variable and g 31 the diagonal represents difference for the two variables [13].

In short, the FMT transformation is a good compromise between the wavelets bases and the diagram of the laplacian pyramid [5].

The continuations f 1 , g 1 can be calculated starting from f 0 in the following way:  1 0 fk,l = f2k,2l    g 11 = f 0 0 0 1   k,l 2k+1,2l − /2(f2k,2l + f2k+2,2l ) 21 0 0 0 gk,l = f2k,2l+1 − 1/2(f2k,2l + f2k,2l+2 )  0 0 31  1/4(f  = f − g  2k,2l 2k+1,2l+1  k,l 0 0 0 +f2k,2l+2 + f2k+2,2l + f2k+2,2l+2 )

The result of the wavelets transformation of an image is represented by a pyramidal sequence of images [7], which includes the differences in information between the successive scales (figure 1). However, we can consider the FMT multi-scale transformation as a linear application, from the canonical base to the multi-scale base, which distributes the information contained in the initial image in a different way. It is thus more natural to visualize this redistribution, in the multi-scale base, in only one image, as it is the case in the canonical base. The principle of the visualization of images in the canonical base consists in placing each coefficient at the place where its basic function reaches its maximum. The same principle is naturally essential for the multi-scale base (Figure 2) [5].

Reciprocally one can rebuild the continuation f 0 from f 1 and g 1 by:  0 1 f2k,2l = fk,l    0 11 1 1   f2k+1,2l = gk,l + 1/2(fk,l + fk+1,l ) 0 21 1 1 f2k,2l+1 = gk,l + 1/2(fk,l + fk,l+1 )  0 31 1  1  f2k+1,2l+1 = gk,l + /4(fk,l   1 1 1 +fk,l+1 + fk+1,l + fk+1,l+1 ) We thus obtain a pyramidal algorithm which, on each scale j, decompose (respectively, reconstructed) the continuation f j in (respectively from) f j+1 and g j+1 . The number of operations used in the algorithm is proportional to the number N of data, which is not invalid in the signal (O(N )) what makes of it a very fast algorithm. What is more, the operations contain only arithmetic numbers; therefore, the transformation is exact and does not produce any approximation in its numerical implementation [5]. The multi-scale transformations were initially introduced into the image processing by the laplacian pyramid algorithm of P. Burt and E. Adelson; then Mallat developed, starting from the orthonormal wavelets bases, a fast algorithm of transformation and reconstruction of the images in these bases [14]. The algorithm of Mallat resembles the algorithm of Burt-Adelson, but it is more effective because it does not generate the redundancy in information. Instead, it eliminates the correlations between the various scales, and gives indications of space orientations in the image [15]. The FMT Transformation has exactly the same principle of construction as that of Mallat except that the canonical base of the multi-resolution analysis is not an orthogonal base. This does not prevent it from having the same properties in image processing as the wavelets bases [8]. In addition, the FMT algorithm is closer to that of the laplacian pyramid, because it is very simple and completely discrete, what makes it possible to observe directly on the pixels the effects of the transformation.

2.1.2 Visualization of the transformed images by the FMT

The image obtained is a coherent one which resembles an outline representation of the original image (Figure 3). Indeed, the FMT transformation, like some wavelets transformation, has similarities with the canny outlines detector, where the outlines correspond to the local maximum in the module of transformation. In fact, in the case of the FMT transformation, on each scale, the value of each pixel is given by the calculation of the difference with its neighboring of the preceding scale. Thus the areas which present a local peak for these differences correspond to a strong luminous transition for the values of grey, while the areas, where those differences are invalid, are associated with an area, where the level of grey is constant [20].

Fig. 1: Representation on separated scales for 9×9 transformed image in the multi-scale base.

Hybrid Methods of Image Compression-Encryption

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2.1.3 Compression of images by FMT

Fig. 2: Representation on mixed scales, the coefficients are placed at the place where their basic functions are maximal.

(a)

(b)

We noted that the FMT transformation of the images gives, by the means of visualization on mixed scales, a good description of the outlines of the image. In fact, the result is more than only one image of the outlines because it contains exactly same information as that of the initial image and we can even find the original image by the reverse transformation. This leads us to raise the question to know if these multi-scale coefficients of the outline areas characterize the image completely. Theoretically, the answer is negative. Indeed, there are, in the case of the continuous wavelets transformations, some counter-examples of different functions which have the same outline points in their wavelet transformation [21]. Hence, we can only wish that the introduced debasement, if we do not hold account of the outline regions, remain unperceivable in the rebuilt image. A worthwhile priority over the FMT transformation, which is also valid for the wavelets transformations, is the characteristic aspect observed in the histograms of transformed images [7]: the number of coefficients for a given level of grey decreases very quickly, to practically fade away, when we move away from any central value very close to zero (figure 4). This implies that the information (or the energy) of the transformed image is concentrated in a small number of significant coefficients, confined in the outline region of the initial image. Therefore, the cancelation of other coefficients (almost faded away) only provokes a small disruption of the transformed image. In order to know the effect of such disruption in the reconstruction of the initial image one should calculate the matrix conditioning of the FMT transformation. In fact, if we have f = Mg where f is the initial image and g is the multi-scale image, then the conditioning of M (Cond(M) = ||M||.||M − 1|| 1) who checks: ||f||/||f|| Cond(M) ||g||/||g||.

(c)

(d)

Fig. 3: Representation on mixed-scales and on separate scales of the image ”Lena”. The coefficients are in the canonical base in (a) and (c) and in the Faber-Schauder multi-scale base in (b) and (d).

This means that the relative variation of the restored image cannot be very important, with reference to the multi-scale image, if the conditioning is closer to 1 [5]. For the orthonormal transformations, the conditioning is always equal to 1; thus it is optimum. However, we can always improve the conditioning if we are able to multiply each column (or each line) by a well chosen scalar; in the case of a base changing, this pushes a change in the normalization of the base elements. The obtained results (Figure 4, 5) confirm that, in this case too, we get a good conditioning. Most generally, we have verified that we can practically eliminate between 90% and 99% of the multi-scale coeffi-

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(a)

(a)

(b)

(b)

(c)

Fig. 5: Degradation by FMT. The percentage of eliminated coefficients: (a) Arches’s original image, (b) 90%, (c) 93.

Fig. 4: Histograms of image Lena : (a) in the canonical base, (b) in the multi-scale base. Table 1: The percentage of eliminated coefficients of images reconstituted. Images

Eliminated coefficients 86% 91% 90% 90%

Arches Echo. 3 Lena Flower

cients, without any remarkable debasement of the reconstructed image, and with a good ratio of noise signal (PSNR) [9]. The results are, obviously preferable, when they are not so textured (table 1). The Mean Square Error (MSE) and the Peak Signal to Noise Ratio (PSNR) are mathematical measures which need the original image, before the compression, in order to measure the distortion [4]. The size of the images is M × N, while the pixels coordinates are (m, n). The MSE measures the square of difference in each point, between the original image and the compressed one [11]: 1 ∑∑ 2 (Iorignal (m, n) −Icompressed (m, n)) . M.N n=1 m=1 M

EQM =

M

The PSNR measures the signal ratio of noise [14]: PSNR = 10log10

Ng2 (en dB) . EQM

Here Ng represents the maximal grey level that a pixel can take. For example, the coded images on 8 bits, Ng = 255 [18]. If we compare the performances of the FMT transformation with the standards method of compression, (JPEG), we will verify that we can reach good results of compression, without debasing the image. What is

more, those results are obtained when applying the multi-scale transformation to the whole image, while the DCT transformation, which is the basis of the JPEG method, is not effective when applied to reduced blocks pixels (generally applied to blocks of size 8 × 8 pixels), what involves the appearance of the blocks of artifacts on the images when the compression ratio is high [13]. This phenomenon of artifacts blocks is not common in the FMT transformation (figure 5). 2.1.4 DES Algorithm The Data Encryption Standard (DES) was jointly developed in 1974 by IBM and the U.S. government (US patent 3,962,539) to set a standard that everyone could use to securely communicate with each other. It operates on blocks of 64 bits using a secret key that is 56 bits long. The original proposal used a secret key that was 64 bits long. It is widely believed that the removal of these 8 bits from the key was done to make it possible for U.S. government agencies to secretly crack messages [1]. DES started out as the ”Lucifer” algorithm developed by IBM. The US National Security Agency (NSA) made several modifications, after which it was adopted as Federal Information Processing Standard (FIPS) standard 46-3 and ANSI standard X3.92 [3]. Encryption of a block of the message takes place in 16 stages or rounds (Figure 6). From the input key, sixteen 48 bit keys are generated, one for each round. In each round, eight so-called S-boxes are used. These S-boxes are fixed in the specification of the standard. Using the S-boxes, groups of six bits are mapped to groups of four bits. The contents of these S-boxes have been determined by the U.S. National Security Agency (NSA). The S-boxes appear to be randomly filled, but this is not the case. Recently it has been discovered that

Hybrid Methods of Image Compression-Encryption

these S-boxes, determined in the 1970s, are resistant against an attack called differential cryptanalysis which was first known in the 1990s [19]. The block of the message is divided into two halves. The right half is expanded from 32 to 48 bits using another fixed table. The result is combined with the sub key for that round using the XOR operation. Using the S-boxes the 48 resulting bits are then transformed again to 32 bits, which are subsequently permutated again using yet another fixed table. This by now thoroughly shuffled right half is now combined with the left half using the XOR operation. In the next round, this combination is used as the new left half [15].

7

The Triple-DES variant was developed after it became clear that DES by itself was too easy to crack. It uses three 56-bit DES keys, giving a total key length of 168 bits. Encryption using Triple-DES is simply (1) Encryption using DES with the first 56-bit key. (2) Decryption using DES with the second 56-bit key. (3) Encryption using DES with the third 56-bit key. Because Triple-DES applies the DES algorithm three times (hence the name), Triple-DES takes three times as long as standard DES. Decryption using Triple-DES is the same as the encryption, except it is executed in reverse. 2.1.5 AES Algorithm

Fig. 6: Diagram of DES Encryption.

The figure should hopefully make this process a bit more clear. In the figure, the left and right halves are denotes as L0 and R0, and in subsequent rounds as L1, R1, L2, R2 and so on. The function f is responsible for all the mappings described above [12]. This secret key encryption algorithm uses a key that is 56 bits, or seven characters long. At the time it was believed that trying out all 72, 057, 594, 037, 927, 936 possible keys (a seven with 16 zeros) would be impossible because computers could not possibly ever become fast enough. In 1998 the Electronic Frontier Foundation (EFF) built a special-purpose machine that could decrypt a message by trying out all possible keys in less than three days.

AES is the acronym of Advanced Encryption Standard, creates by Johan Daemen and Vincent Rijmen. It is a technique of encryption to symmetrical key. It is the result of a call to world contribution for the definition of an algorithm of encryption, call resulting from the national institute of the standards and technology of the government American (NIST) in 1997 and finished in 2001. this algorithm provides a strong encryption and was selected by the NIST like normalizes federal for the data processing (Federal Information Processing Standard) in November 2001 (FISP-197), then in June 2003, the American government (NSA) announced that AES was sufficiently protected to protect the information classified up to the level TOP SECRET, which is the most level of safety defined for information which could cause “exceptionally serious damage” in the event of revelations with the public [2]. Algorithm AES uses one the three lengths of key of coding (password) following: 128, 192 or 256. Each size of key of encryption uses a slightly different algorithm, thus the higher sizes of key offer not only one greater number of bits of jamming of the data but also an increased complexity of the algorithm [1]. This algorithm always preserves the high level of safety proposed by DES; indeed the process is always based on a function of expansion E, boxes of substitutions S, called on the level of the diversification of key K; moreover, the process always preserves the principle of the stages at the moment of the stage of expansion. The innovation brought by the AES is noted on the level of the size of the secret key as well as the size of the data treated in entry; precisely we pass from a key of coding of size 64 bits (8 bytes) for the case of worms a key of size doubles 128 bits (16 bytes) for the AES. The size of the data with crypter is as notably larger as that of since we pass from 64 bits towards 128 bits [16]. Moreover, as for DES, the AES is a cryptographic

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system with secret key; what makes the operation of Encrypting-decrypting rather light [3]. The size of the data treated by the AES (16 bytes) gives us the possibility well of exploiting the supporting algorithm in applications of the data files of large size [23]. Cryptography with symmetrical algorithms uses the same key for the processes of Encrypting and Decrypting; this key is generally called ”secret” (in opposition to ”private”) because all the safety of the unit is directly related to the fact that this key is known only by the shipper and the recipient [23]. Symmetrical cryptography is very much used and is characterized by a great speed (encrypting with the flight, ”one-the-fly”), implementations as well software (Krypto Zone, firewalls software Firewall-1 type and VPN-1 of Checkpoint) that hardware (dedicated charts, processors cryptos 8 to 32 bits, . . . ) what accelerates the flows clearly and authorizes its massive use. This type of cryptography usually functions according to two different processes, encrypting per blocks and the encrypting of ”stream” Fig. 7: Diagram of AES algorithm, version 128 bits. (uninterrupted). Algorithm AES is iterative (Fig. 7). It can be cut out in 3 blocks [23]: (I) Initial Round. It is the first and the simplest of the stages. It counts only one operation: Add Key Round. (II) N Rounds. N is the iteration count. This number varies according to the size of the key used. 128 bits for N = 9, 192 bits for N = 11, 256 bits for N = 13. This second stage consists of N iterations comprising each one the four following operations: Sub Bytes, Rows Shift, Mix Columns, add Key Round. (III) Final Round. This stage is almost identical to the one of the N iterations of the second stage. The only difference is that it does not comprise the operation Mix Columns.

2.2 Principle schema of proposed compression-encryption approaches The essential idea is to combine the compression and the encryption during the procedure. It is thus a question of immediately applying the encryption to the coefficients of the preserved compression, after the application of transformed FMT to visualization in mixed scales. General diagram is given on figure 8 as follow: It consists in carrying out an encryption after the stage of quantization and right before the stage of entropic coding. To restore the starting information, one decodes initially the quantified coefficients of the FMT matrix

Fig. 8: General diagram of proposed compressionencryption approaches.

by the entropic decoder. Then, one deciphers them before the stage of quantization. Lastly, one applies the IFMT (reverse FMT) to restore the image. The principal advantages of these approaches are the flexibility and the reduction of the processing time during the coding and decoding operations. Indeed, by

Hybrid Methods of Image Compression-Encryption

9

these methods, one can vary the processing time according to the desired degree of safety.

3 Results 3.1 Applications The results obtained after the application of methods FMT-AES and FMT-DES on the images (placeLena), (Image Echo.3), (Flower), (arches), (Image Echo.1) and (Image Echo.2) are given as follows: (a)

(b)

Fig. 10: Histograms of originals images and histograms of compressed and encrypted images by FMT-AES.

(a)

(b)

(a)

(b)

Fig. 11: Stages after application of FMT-DES method on echographic images.

3.2 Comparison

(c)

(d)

Fig. 9: The stages after application of FMT-AES method on Lena’s image, Echographic image, Flower’s image and Arches’s image.

The comparison is carried out, after the application of the methods of compression-encryption: DCT-RSA, Quadtree-AES, DCT-partial encryption, FMT-DES, and FMT-AES, on the image ”placeLena”, ”Flower”, ”Arches” and ”echographic images: Echo.1, Echo.2 and Echo.3”. It should be noted that the resolution of the images is 256×256 dpi, and the processor used is Intel Pentium4 for a rate equalizes 3.46 GHz. The results obtained are given on table 2 following: To apply the encryption methods; RSA and Partial encryption, we propose to quantify only the frequential coefficients relating to the low frequencies. By

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Lena Flower Arches Echo.1 Echo.2 Echo.3

7.689 9.085 8.880 9.151 11.285 8.451

Q.O.I

0.92 094 0.93 0.88 0.86 0.85

Quadtree-AES PSNR E.R.I. (dB) 35.201 7.189 32.605 8.585 46.886 8.984 43.771 7.185 41.991 8.581 41.316 8.013

DCT-partial encryption PSNR E.R.I. Q.R.I. (dB) 35.728 7.231 0.88 33.136 8.626 0.90 44.643 8.223 0.86 43.918 7.234 0.83 41.459 8.623 0.82 41.855 7.960 0.82

Fig. 12: Histograms of originals images and histograms of compressed and encrypted images by FMT-DES. E.O.I.

Lena Flower Arches Echo.1 Echo.2 Echo.3

7.689 9.085 8.880 9.151 11.285 8.451

FMT-DES PSNR E.R.I (dB) 34.531 7.104 31.933 8.500 43.444 8.094 42.730 7.107 40.261 8.499 40.651 7.981

Q.O.I.

0.92 094 0.93 0.88 0.86 0.85

Q.R.I 0.88 0.90 0.88 0.83 0.82 0.82

DCT- RSA PSNR E.R.I. (dB) 35.401 7.181 32.803 8.576 44.317 8.169 43.603 7.181 41.131 8.575 41.529 7.905

Q.R.I. 0.89 0.91 0.89 0.85 0.83 0.82

FMT-AES C.T. PSNR E.R.I (s) (dB) 7.47 34.533 7.103 8.56 31.933 8.501 7.71 43.445 8.093 7.02 42.731 7.109 6.90 40.263 8.498 7.34 41.001 7.785

C.T. (s) 13.95 15.03 14.25 13.51 13.40 13.82

Q.R.I C.T. (s) 0.88 3.15 0.90 4.26 0.88 3.37 0.83 3.09 0.82 3.19 0.82 3.01

quantifying all the coefficients of the first column and the first line of the blocks 8×8, the size of the cryptocompressed image is closer to the size of the original image. In this case we lose in compression ratio. It should be noted that the DCT-RSA, DCT-Partial encryption and FMT-DES methods require a long processing time, while these methods depend on the coefficients selected before the realization of the encryption. DCT-RSA and DCT-Partial encryption lead to the appearance of the artifact blocks on the reconstituted images when the compression ratio is high. This Phenomenon of artifact blocks is not known any more in

Q.R.I. 0.88 0.90 0.89 0.83 0.82 0.83

C.T. (s) 4.30 4.59 5.83 4.23 4.41 4.17

C.T. (s) 9.92 11.01 10.11 9.87 10.01 9.83

Table 2: Comparison of methods ”DCT-RSA”, ”QUADTREE-AES”, ”DCT-PARTIAL ENCRYPTION”, ”FMT-DES” and ”FMT-AES”. E.O.I.: entropy of original image, Q.O.I.: quality of original image, E.R.I.: entropy of reconstituted image, Q.R.I.: quality of reconstituted image, C.T.: computing time.

the FMT transformation. For the DCT-RSA encryption method, we kept the coefficients of the first line and the first column, after the application of the DCT transformation on each block of 8×8 pixels. In general, we tried to have same quality by applying the various methods of encryption-compression. The principal advantages of our two approaches: FMT-DES and FMT-AES, are the flexibility and the reduction of the processing time, which is proportional to the number of the dominant coefficients, at the time of the operations of encryption and decryption. Indeed, by our two methods, one can vary the processing time according to the desired degree of safety.

4 Conclusion We presented two approaches of compression-encryption which are based on the Faber-Schauder Multi-scale Transformation, stemming from the expression of the images in the Faber-Schauder base and the two encryption algorithms: DES and AES. The FMT transformation is distinguished by its simplicity and its performances of seclusion of the information in the outline regions of the image. The mixed-scale visualization of the transformed images allows putting in evidence its properties, particularly, the possibilities of compression of the images and the improvement of the performances of the

Hybrid Methods of Image Compression-Encryption

other standard methods of compression as JPEG and GIF. AES encryption algorithm leaves, in the stage of compression, homogeneous zones in the high frequencies. It is approximately twice faster to calculate (in software) and approximately 1022 times surer (in theory) that DES. However, even if it is easy to calculate, it is not it enough to be taken into account in the current Wi-Fi charts. The standard 802.11i will thus require a renewal of the material to be able to make safe the networks of transmissions without wire. The comparison of FMT-AES method with the methods: FMT-DES, Quadtree-AES, DCT-partial encryption and DCT-RSA, showed well its good performance. Finally, we think of using hybrid methods in compression and encryption by mixture of data and setting up a encrypt analysis of the proposed approaches.

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