DOUBLE IMAGE ENCRYPTION BASED ON THE RECIPROCAL-ORTHOGONAL PARAMETRIC TRANSFORM AND CHAOTIC MAPS

Seif Eddine Azoug and Saad Bouguezel

Laboratoire de Croissance et Caractérisation de Nouveaux Semi-conducteurs (LCCNS)

Department of Electronics, Faculty of Technology, Ferhat Abbas University Setif -1, Setif, Algeria E-mails: azoug.se@gmail.com, bouguezel_saad@yahoo.com

ABSTRACT

In this paper, we propose a double image encryption method based on the reciprocal-orthogonal parametric (ROP) transform and chaotic maps. In this method, a complex-valued image is constructed by two secret real-valued images, one as amplitude and the other as phase. In addition, two chaotic random phase masks are generated using two non-independent chaotic maps; one mask is multiplied by the resulting complex-valued image before applying the two-dimensional (2-D) ROP transform and the other one is multiplied by the resulting matrix in the transform domain. This step is then followed by a chaotic scrambling between the real and imaginary parts before applying another 2-D ROP transform, which yields the encrypted image. The independent parameters of the ROP transforms and the parameters of the chaotic maps used for the masks and scrambling are successfully exploited as an encryption secret key. Simulation results demonstrate the robustness of the proposed method against blind decryption, brute force and statistical attacks.

1. INTRODUCTION

Due to the widespread use of multimedia technology in our daily life, the multimedia encryption field, particularly image encryption, is becoming important to keep privacy and security of sensitive images in military, medical and commercial sectors. Several encryption methods have been developed in the literature based on different approaches [1]-[5]. Among them, the well-known double random phase encoding (DRPE) technique [2], in which a secret image is converted to a stationary white noise by multiplying it element-by-element with a random phase mask (RPM) in the spatial domain and with another RPM in the Fourier domain. The phases of these masks are independent and uniformly distributed in the interval [0, 2 ] and the RPM in the transform domain is considered as the main encryption key. To enhance the encryption key space, the DRPE technique has been expanded to the fractional Fourier domain and other

parametric transform domains [3]-[5], amongst them the reciprocal-orthogonal parametric (ROP) transform. All these techniques encrypt only a single real-valued image leading to a complex-valued image. This can be a drawback in terms of storage and/or transmission load, which can be solved, in some applications, by encrypting two real-valued images at the same time. To deal with this issue, several methods based on the DRPE technique and parametric transforms for double image encryption have been proposed in [6]-[9]. In these methods, generally two secret real-valued images are encoded into one complex-valued image before applying the DRPE technique. Moreover, a scrambling function is introduced either in the spatial domain or frequency domain between the elements of the real and/or imaginary parts of a complex-valued image. Some methods use chaotic maps as a scrambling function [9]. The chaotic maps are well-known by their Gaussian like correlation properties and high dependency to their initial parameter [10], which is known also as the seed value. In addition, the chaos encryption has shown its efficiency within the DRPE technique for single image encryption [11]-[13], where the RPM is substituted in [12] by the chaotic random phase mask (CRPM). One of the main advantages of using CRPM instead of RPM in the DRPE technique is to have a flexible secret key possessing high sensitivity to errors.

In this paper, we propose a double image encryption method based on the ROP transform and chaotic maps. Two secret real-valued images are first encoded as amplitude and phase of a complex-valued matrix without scrambling in the spatial domain, and then two CRPM are generated such as the seed value of the second CRPM depends on that of the first CRPM. The next step is to multiply the encoded complex-valued matrix element-by-element with the first CRPM before applying the ROP transform, which is based on the well-known Hadamard matrix and has a simple and regular structure, low complexity compared to the fractional Fourier based-transforms and /2 − 1 independent parameters that can be chosen arbitrarily from the complex plane [14] and exploited as an additional encryption key [5]. The matrix obtained after transformation is multiplied element-by-element with the second CRPM in the ROP domain, in which a chaotic scrambling between the real and

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2013 8th International Workshop on Systems, Signal Processing and their Applications (WoSSPA)

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imaginary parts of the resulting matrix is performed before applying another ROP transform to obtain the final encrypted image. The incorporation of the chaotic scrambling in the ROP transform domain enhances the encryption key space. Finally, the secret encryption key for the proposed method consists of the first CRPM seed value, the parameters of the ROP transforms, the control parameters of the second CRPM and the scrambling chaotic map parameters.

The paper is organized as: in Sections 2 and 3, we briefly review the ROP transform and the chaotic maps within the proposed chaotic scrambling, respectively. In Section 4, we propose a double image encryption method by exploiting the ROP transform and the chaotic scrambling in the ROP domain. Simulation results and performance analysis are presented in Section 5, and finally, a conclusion is included in Section 6.

2. RECIPROCAL-ORTHOGONAL PARAMETRIC TRANSFORM

In this section, we briefly review the ROP transform defined in [14].

Let a symmetric vector of length = 2 , where is a positive integer, defined as = 1 … … 1

(1)

with , = 1, 2, …, /2 − 1, are non-zero complex parameters. Let be an × Hadamard matrix whose rows are indexed by a variable , 0 ≤ ≤ − 1. The binary decomposition of n is given by = 2 + 2 + ⋯ + 2 + (2)

where the value of the binary digit , 0 ≤ ≤ − 1, is 0 or 1. A row of the Hadamard matrix indexed by , is called minus indexed row if satisfies (−1)∑ = −1.

The construction of the ROP matrix of order is realized by performing an element-by-element multiplication of each minus indexed row of the Hadamard matrix by the parametric vector . This vector has ( /2) − 1 independent parameters and is called the parametric vector of the matrix . The resulting matrix is ROP, i.e., . = , where is the identity matrix and (. ) denotes the reciprocal and transpose operations, knowing that the reciprocal operation on a given matrix consists of taking the reciprocal of each of the entries without changing

their locations (. ) = ((. ) ) = ((. ) ) . The ROP transform has some interesting properties [14] such as simple and regular structure, low complexity and possession of ( /2) − 1 independent parameters that can be chosen arbitrarily from the complex plane. Explicitly, these parameters can be exploited as an additional encryption key advantaging the encryption key space.

3. CHAOTIC MAPS

A chaotic map is a non-linear set of values generated iteratively { , = 0,1,2,3, … } in the interval [0,1], this maps has Gaussian like correlation properties, ergodicity, pseudo-randomness, and extremely sensitive to the initial condition [10],[11] , also known as the seed value [12]. A minor error in the seed value results in a completely different set of values. In [12], different chaotic maps are proposed to generate CRPM with a uniform distribution such as the RPM in the DRPE. The authors compared three kinds of chaotic maps, specifically, the logistic map, the tent map and the Kaplan-Yorke map. This last map is a two dimensional map and is the suggested map in [12] for CRPM construction, however, in our method, we exploit the one dimensional logistic map instead of the two dimensional Kaplan-Yorke map because of its simplicity. The logistic map is an iterative quadratic map defined in iterative form as

= . . (1 − ) (3) ∈ [0,1] , 3.99456 ≤ ≤ 4

where is the control parameter , ∈ [0,4] but if 3.99456 ≤ ≤ 4 we get a full nonlinear map, however, it can be ≅ 4 based on the bifurcation plot [11], is the seed value, and are the real iterative values generated by (3). In our work, we use two logistic maps to generate two CRPM in a way similar to that in [12], but the seed value of the second logistic map of the second CRPM is equal to the last iterative value of the logistic map used to generate the first CRPM, this makes a dependency between both CRPM and enhance double image encryption security, hence, an error in the first CRPM will spread to the second CRPM influencing both the amplitude-based and phase-based encrypted images. A third logistic map is generated with parameters ( , ) to perform a chaotic scrambling in the ROP domain. For a complex-valued image matrix of size × . A chaotic scrambling denoted by { , }(. ) is performed as follow:

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ʘ { , }(. )

{ , }(. )

| . | 1π arg(. )

{ , } { , }

, ∗

Inverse 2D ROP ( , )

Inverse 2D ROP ( , )

Forward 2D ROP ( , )

Forward 2D ROP ( , )

exp ( . )

Figure 1. Proposed double image encryption/decryption

, ∗

ʘ ʘ

ʘ ʘ

1) Generate iteratively the elements of a vector of size × using the logistic map in (3) with the parameters ( , ), and then sort the obtained values in an ascending order to create a vector . The permutations index representing the sorting order are recorded in a vector such as ( ) = ( ) , = 1,2,3, … , × (4)

2) The matrix is reshaped into a complex-valued vector of size × × 1 , and then the obtained vector is fragmented into real elements vector { ( ), = 1,2,3, … , × } and imaginary elements vector { ( ), = 1,2,3, … , × }. The parts of this two vectors are shuffled between each them using the permutation index vector to form the shuffled vectors ′ and ′, respectively, such as ′ ( ) = ( )′( ) = ( ) , = 1,2,3, … , × (5)

3) The two vectors ′ and ′ are merged to form a

complex vector of size × × 1 such as + ′ where = √−1, , and then this vector is reshaped into a complex matrix of size × . This matrix is the complex-valued matrix chaotically scrambled.

The inverse of the chaotic scrambling is denoted by { , }(. ), where we obtain the permutation index vector ′ similarly to the Step 1 . The scrambled matrix

is reshaped into × × 1 vector, and the complex vectors elements are shuffled as in Step 2 with the permutation index vector ′. The final shuffled vector is converted to a complex vector before reshaping it to a matrix of size × as in Step 3 to obtain the original

matrix . The seed value and control parameter of the chaotic map are both considered as encryption key.

4. PROPOSED DOUBLE IMAGE

ENCRYPTION/DECRYPTION

In this section, we propose a double image encryption and decryption in the ROP domain as shown in Fig. 1.

Let consider two real-valued square images matrices and of the same size × regarded as the

amplitude and phase of one complex-valued matrix and encoded as

= . exp ( . . ) (6)

where = | | and = arg( ) . Two CRPM { , } and { , } of size × are generated using two logistic maps vectors each one of size using equation (3) with parameters ( , = 4) and ( , ), respectively. Note that the seed value of the second logistic map is equal to the last iterative value of the first logistic map. The two obtained maps are then reshaped into matrices of size × , and each matrix is encoded with the phase function ( 2 . ) to form CRPM { , } and { , }. The next step is to multiply the previous complex matrix element-by-element with the mask { , } before applying a 2-D ROP transform in a row-column fashion on the resulting matrix using the ROP matrices and with the parametric vectors and , respectively. The obtained matrix in the ROP domain is then multiplied element-by-element with the mask { , } , and the resulting matrix is scrambled as described in the previous section using the chaotic scrambling function { , }(. ) with the parameters ( , ), and lastly, the final encrypted complex-valued image is obtained by applying another 2-D ROP transform on the chaotic scrambled matrix by

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using the ROP matrices and in a row-column fashion to obtain the final encrypted image such as = { , } ⊙ { , } ⊙

, (7)

where ⊙ denotes the element-by-element multiplication operation. Note that each of the vectors , , , and

has /2 − 1 independent parameters that can be chosen arbitrarily from the complex plane. In summary, the encrypted image is obtained using 4( /2 − 1) +4 = 2 parameters as the encryption secret key, these parameters are the parameters used to generate the CRPM

and respectively, , , the parameters ( , ) used in the chaotic scrambling and the 4( /2 − 1) independent parameters of the four ROP matrices. The parameters and must be chosen ≅ 4 to approach the full non linearity map of the logistic map.

The decryption process is the reverse of the above encryption process and can be summarized as = , ⊙

, ∗ ⊙ , ∗ (8)

where (. )∗ denotes the complex conjugate, , = 1, 2, 3 and 4, are parametric vectors each has /2 − 1 independent parameters that can be chosen arbitrarily from the complex plane, and ( , = 4) , , and ( , ) are the chaotic parameters used in the decryption process. Finally, the phase-based image and the amplitude-based image are decoded by taking = | | and = arg( ) . It is clear from (6)-(8) that the decrypted/decoded images and become identical to their respective original images only when = , = , = , = 1, 2, 3 and 4, and also when = , and = .

5. SIMULATION RESULTS

In this section, we implement the proposed double image encryption and decryption method presented in the previous section using various standard test images to show its efficiency and robustness. The following parameters values ( , ) , and are chosen respectively (0.541,4) , and 3.9 to generate the CRPM { , } and { , }, additionally, parameters ( , ) are chosen as ( 0.371,3.983 ), however, the 2 − 4 independent parameters of the vectors , = 1, 2, 3 and 4, of the four ROP transform matrices , = 1, 2, 3 and 4, are arbitrarily chosen from the complex plane. Notice that the secret real-valued images and , the masks { , }

and { , }, and the ROP matrices , = 1, 2, 3 and 4 are all of square size × .

If we consider the case of 512 × 512 gray scale input image, for boat image and cameraman image , the simulation results are shown in Fig. 2. The Fig. 2(a), and Fig. 2(b) present the original boat image, and cameraman image respectively, and the Fig. 2(c), and Fig. 2(d) correspond respectively, to the amplitude, and the phase of the encrypted image. The decrypted images using incorrect values for all the decryption parameters are shown in Fig. 2(e) and Fig. 2(f). Fig. 2(g) and Fig. 2(h)

(a) (b)

(d) (c)

(e) (f)

(h) (g)

Figure 2. Results of the proposed double image encryption method: (a), (b) original images, (c), (d) amplitude and phase of the encrypted images (e), (f) decrypted images with wrong

keys, (g), (h) decrypted images with correct keys.

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(c) (d)

(e) (f)

Figure 4. Decrypted images with: (a), (b) all parametric vectors are wrong, (c), (d) [ = ] ,and (e), (f) [ =, = , = ]

(a) (b)

show the decrypted image when the decryption parameters are correct.

In order to demonstrate the degree of sensitivity to

errors in the parameters of the chaotic maps used for scrambling and CRPM generation, let assume in the decryption process that the four ROP parametric vectors are correct, i.e. = , = 1, 2, 3 and 4, and one of the parameters , , and ( , ) are marginally different from those used in the encryption process. The decrypted images with = + 10 are shown in Fig. 3(a) and Fig. 3(b), respectively, and with = + 10 the decrypted images are shown in Fig. 3(c) and Fig. 3(d). In addition, to show the sensitivity of our method to the

chaotic scrambling, the decrypted images with the following chaotic scrambling parameters ( =+10 , = ) are shown in Fig. 3(e) and Fig. 3(f), and with the parameters ( = , = + 10 ) results are shown in Fig. 3(g) and Fig. 3(h). These results confirm the encryption key sensitivity to errors and the robustness of the proposed method against blind decryption. These show also that the use of the control parameters of the chaotic maps in the transform domain enhanced the encryption key space and sensitivity.

To show the robustness of the proposed method

against brute force attacks, let analyze the effect of the 2 − 4 independent parameters of the vectors , = 1, 2, 3 and 4, of the four ROP transform matrices , = 1, 2, 3 and 4. For this purpose, we assume in the following simulations that all the chaotic maps parameters in the decryption process are correct. The decrypted images shown in both Figs. 4(a), (b), and Figs. 4(c), (d), and Figs. 4(e), (f) are obtained with, [ ≠ , ≠, ≠ , ≠ ],[ = , ≠ , ≠, ≠ ], and [ = , = , = , ≠], respectively. It is clear from these figures that the decrypted image is sensitive too to the errors in the

(a) (b)

(d) (c)

(e) (f)

(h) (g)

Figure 3. Decrypted images with: (a), (b) = +10 , (c), (d) = + 10 ,(e), (f) = +10 , and (g), (h) = +10

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(b) (a)

Figure 5. (a) Amplitude histogram and, (b) phase histogram of the encrypted complex-valued image

independent parameters of the ROP transform. All the previous results confirm the robustness of the proposed method against brute force attacks due to the being of the chaotic maps in joint with the ROP transforms parameters in the transform domain, hence, making the encryption key space larger.

Furthermore, let analyze the security of the proposed method against statistical attacks, but due to lack of space, only the histogram results of the encrypted images of Boat and Cameraman are shown in Fig. 5(a) and Fig. 5(b). Same results are found for Mandrill and Elaine images of the same size. It is clear that the distribution of the amplitude histogram and distribution of the phase histogram of the encrypted images are completely different from the original images. This confirms the robustness of the proposed method against statistical attacks.

6. CONCLUSION

In this paper, an efficient double image encryption method based on the ROP transform and chaotic maps has been proposed. The simulations carried out show clearly the robustness of the proposed method. It has been shown by testing the sensitivity of the proposed method to different marginal errors in the encryption key, including the control parameters of the chaotic maps, that the chaotic scrambling in the transform domain , the use of the ROP transform parameters, and the dependency between the CRPM enhance the encryption key space and sensitivity in the double image encryption, thus, confirm efficiency against blind decryption and brute force attacks, where the parameters of the chaotic maps of the CRPM and the chaotic scrambling parameters joined with the independent parameters of the ROP transforms can successfully be used as the encryption secret key. It has also been shown by comparing the histograms of the encrypted images with their original images that the proposed method is robust against statistical attacks.

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