A New Multistage Multiple Image Encryption using a combination of Chaotic Block Cipher and Iterative

Fractional Fourier Transform Deepak.Ma*, Ashwin.Va, Amutha.Ra

a Department of Electronics and Communication Engineering, SSN College of Engineering, Chennai, India. * Corresponding author: deepakmuralidharan2308@gmail.com Contact No: +91 9003264278

Abstract — In this work, a new Multistage multiple image encryption using a combination of a 2-round Chaotic Block Cipher and Iterative Fractional Fourier Transform method is proposed. The Chaotic Block Cipher uses a logistic map to generate chaotic numbers and the Iterative Fractional Fourier Transform block is used to combine two images with phase of the images and the fractional orders as keys. The proposed algorithm consists of N stages to encrypt N different input images into a single final encrypted image. The performance of the proposed algorithm is evaluated using various types of analysis such as histogram analysis, correlation analysis, mean square error analysis and key sensitivity analysis. The results indicate that the proposed algorithm is highly secure and occupies less space while transmitting the encrypted data through the network as all the N images are encrypted into a single image of the same size as that of the input images.

I. INTRODUCTION Image encryption has been one of the most widely used techniques in the past decade for securing the confidentiality of the image while transmitting through a network. In recent times, image encryption using chaotic maps has become a popular area of research due to its high sensitivity to initial conditions and the known system parameters [1] and many chaotic based algorithms have been proposed. An extended substitution-diffusion based system using chaotic standards and logistic map was presented in [2] to overcome the disadvantages of the traditional chaotic encryption system. Improving on this method, a new chaotic block cipher based on the S-box algorithm and a combination of the Logistic and the Baker map was proposed to achieve balance in energy consumption and security [3]. But, this algorithm uses more memory space and computation time. Thus, a fast chaotic block cipher for image encryption using piecewise linear chaotic map was proposed to reduce memory space, computation time and improve security by using large sized keys [4]. But these algorithms have primarily been used for encryption of a single image. Multiple image encryption has been an important area of research and in the past few years many algorithms pertaining to this have been proposed. Recently, a double image encryption technique was proposed to combine two images in their Fractional Fourier domains [11]. However, this algorithm becomes less efficient for more than two images as the interval between the phases reduces with the increased number of images which results in an inefficient decryption due to the phase dependent keys and also becomes more susceptible to

attacks. Several multiple image encryption techniques using random phase filtering techniques [8], [9], [10] and cascaded fractional Fourier transforms have also been proposed in the last decade [14]. However, the encrypted images obtained are not distributed with equal probability and hence vulnerable to attacks. Most of the algorithms mentioned above use a phase retrieval technique whereas [5] proposed a double encryption technique by converting the two images using a VSS technique by constructing random grids. A positional multiplexing technique in the Fresnel-transform domain using a modified version of the Gerchberg-Saxton algorithm was proposed in [6]. Furthermore, a chaotic map algorithm utilizing the discrete wavelet transform and the finite Ridgelet transform was suggested in [7]. One inherent disadvantage of this algorithm is that the encrypted image is double the size of the original image and hence occupies more memory space while transmitting through the network. To overcome these drawbacks, a novel algorithm for a multi-stage multiple image encryption technique using a combination of a 2-round chaotic block cipher (CBC) and an Iterative fractional Fourier Transform (IFF) block is proposed in this paper. In the first stage of our method, the original image is sent through the chaotic block cipher [4]. The encrypted image from the first stage is combined iteratively with the second input image in their fractional Fourier domains using the double image encryption technique and another encrypted image is obtained. This is once again sent through the 2-round CBC. This process is repeated consecutively for N stages to encrypt the N original images to form a final single encrypted image. The remainder of this paper is organized as follows. In Section II, our proposed multiple image encryption algorithm is explained in detail. The numerical results of our simulations and the performance analysis are discussed in Section III and finally Section IV concludes the paper.

II. OUR PROPOSED ALGORITHM In our proposed algorithm, the encryption of multiple images is performed using chaotic map and Fractional Fourier Transform (FrFT). Each stage consists of an IFF block followed by a 2-round CBC block. The number of stages N is equal to the number of images to be encrypted. The first stage consists of only the 2-round CBC. Three stages were considered for our simulation purposes (N=3). However, this

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can be extended to any value of N. The first image is divided into sub-images of dimension 8×8 pixels and two 128 bit keys X1 and X2 are chosen. The chaotic numbers c1 and c2 for the first sub-image are generated using the discrete Logistic maps given by xn+1 = xn. .(1-xn) (1) The digitized values s and t are evaluated as s = 2. E(2p.|c1|) (2) t = 2. E(2p.|c2|) (3) The Linear Diophantine Equation (LDE) given by (4) is solved with the digitized chaotic numbers as coefficients. sx + ty = c (4)

The solutions of the LDE x and y are used to generate the permutation key IP given as IP=IQ (IR) (5) Here, IQ and IR are the sorted index of the LDE solutions x and y respectively. For example, x=(q(1),q(2),….,q(n)) and y=(r(1),r(2),….,r(n)) where n is the permutation length. The solutions x and y are sorted either in ascending or descending order and the index of resultant x and y is given as IQ and IR respectively. The shuffled sub-image W using the permutation key IP is evaluated as W=S (IP) (6) where S is the sub-image. W is diffused using the diffusion key ID to obtain the encrypted sub-image. New chaotic numbers are generated using Eq.(1) and the permutation key IP and diffusion key ID are refreshed for the next sub-image. With the new chaotic numbers and keys, the next sub-image is encrypted. This process is continued till all the sub-images are encrypted. Now with the encrypted image and new chaotic values the entire process i.e. from Eq.(2)-Eq.(6) is carried out to get the final encrypted image. The final encrypted image of the 2-round CBC and the new image are given as inputs to the IFF block. A and B are assumed to be the sub images of the final encrypted image and new image respectively of dimensions 16×16 pixels. An initial phase matrix 1 is considered. The sub-image A and phase 1 are combined in the fractional Fourier Domain of order = –

using the equation: FrFT (A. exp (1i× 1)) = A2. exp (1i× 2) (7) The amplitude is ignored and the phase of the resultant matrix is extracted and combined with the second input image equation defined as

FrFT- (B. exp (1i× 2)) = C. exp (1i× c) (8) With the values of 1 and c the repeat phase matrix repeat is obtained using the function defined as 1, if amplitude (A2) amplitude (B)

repeat = 2 c - 1, if amplitude (A2) > amplitude (B) (9) For the next iteration, the same steps mentioned above are repeated but with the phase matrix replaced as . This is repeated until the value of approaches . The final encrypted image E can be obtained as E.exp (1i× e) = FrFT (A. exp (1i× 1)) (10) The above steps are repeated till all the sub-images are encrypted. The amplitude of the encrypted image of the IFF block is given as input to the 2-round CBC to generate the encrypted image of the second stage. For the third stage, the encrypted image from the second stage and the third new image are combined in the Fractional Fourier domain to obtain a new encrypted image. The new encrypted image is once again given to the chaotic block cipher to obtain the final encrypted image. Hence, for N images to be encrypted a 2-round CBC followed by (N-1) stages which comprises of a combination of the IFF and the 2-round CBC is required.The block diagram of our algorithm is shown in Fig.1. The pseudo codes of the algorithms for the 2-round CBC and IFF blocks are given below. Pseudo code 1: 2-round chaotic block cipher

1) Divide the image into sub-images of size 8×8 and choose two 128 bits keys X1 and X2.

2) Using the discrete Logistic map equation given in Eq.(1) obtain the chaotic values of c1 and c2.

3) Digitize the values c1 and c2 using Eq.(2) and Eq.(3). 4) Solve the LDE given in Eq.(4) to get the values of x

and y. 5) Obtain the permutation key IP=IQ (IR), where IQ and IR

are the sorted index of LDE solutions. 6) Obtain the shuffled sub-image using the key IP and

diffuse the pixels with diffusion key ID. 7) Increment sub-image, find out new chaotic numbers

and refresh keys IP and IQ. 8) Repeat the steps 6 and 7 until all sub-images are

encrypted. 9) Generate new chaotic values c1 and c2 using Eq.(1)

and perform steps 3-8 for the encrypted image obtained in the first round.

Pseudo code 2: Iterative fractional Fourier Transform block

1) Assume an initial phase 1. 2) Combine the input image A and phase 1 in the

fractional Fourier Domain of order = – using Eq.(7).

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3) Neglect the amplitude A2 and extract the phase 2 of the resultant matrix and combine it with the second input image B in the fractional Fourier Domain of order – as mentioned in Eq.(8).

4) Obtain the repeat phase matrix using Eq. (9). 5) In the next iteration, replace the phase matrix 1 with

repeat and perform steps 2-4 until A2 nearly approaches B.

6) The final encrypted image can be obtained using Eq.(10).

7) Repeat steps 1-6 till all the sub-images are encrypted.

III. NUMERICAL RESULTS AND PERFORMANCE ANALYSIS

The three original images that were considered for our simulation are illustrated in Fig.2 and the images were encrypted using the algorithm mentioned in Section II. For the 2-round CBC, the initial values x0 and (in Eq.(1)) were assumed to be 0.3521 and 3.9999 respectively. Based on the initial conditions, a chaotic sequence was generated and the encryption algorithm for each sub-block of size 8x8 was performed. For the IFF block, the discrete fractional Fourier Transform algorithm mentioned in [13] was used to compute the FrFTs. The initial phase 1 was assumed to be and 1200 iterations were performed until the amplitudes of A2 and B converged. Also, since one of the constraints of the algorithm is that the Parseval's energy theorem needs to be satisfied [11], the value of a constant k was found out using the Dichotomy's algorithm [12] by comparing the energies of the two images and this constant k was added to the second image to equalize the energies for efficient computation. Also for our computation, the values of and were set to be 0.7 and 0.4 respectively. In order to evaluate the effectiveness of our algorithm and prove that our method is highly efficient and secure to attacks, various tests such as the mean square error analysis, histogram analysis, the correlation analysis and the key sensitivity analysis were performed which is explained in the following sub-sections.

A. Mean Square Error Analysis Fig.2 shows the original images (a-c) and the decrypted images (d-f) obtained for the three stages of our algorithm and it is well evident from the results that the images are decrypted with maximum effectiveness. Furthermore, the mean square error value between the original and the decrypted image retrieved at every stage was calculated to the formula:

(11) where u is the decrypted image, z is the original image and P and Q are the dimensions of the image. For all stages, the errors were found to be very minimal (MSE<100). Hence this

Fig.1. Our proposed encryption algorithm (for N=3) proves that the entire information of the original image is retained in the decrypted image. B. Histogram Analysis In Fig.3, the encrypted images obtained at every stage of the algorithm are displayed along with their histograms. It can be seen that the encrypted images obtained are distributed with equal probability and hence it can be concluded that the information at every stage is extremely resistant to attack. Also, the entropy of the ciphered image at every stage was found to be close to 8 which means that the encryption is very complex adding to the increased security. C. Correlation Analysis

The correlation between adjacent pixels for the original and encrypted images for the three stages is shown in Table I. The correlation between the adjacent pixels of the original images were found to be very high whereas, for the encrypted images, the correlation was found to be very low (nearly 0) due to the increased randomness of the ciphered image.

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Fig.2. (a-c) represent the original images (Lena, bird, camera) of the three stages and (d-f) represent the decrypted images. . D. Key Sensitivity Analysis In order for an algorithm to be considered effective, it must be sensitive to even small changes in the key. In order to test the key sensitivity, the least significant bit (LSB) and the most significant bit (MSB) of the 128-bit keys X1 and X2 was changed. The results showed that our algorithm is very sensitive to even the smallest key change and hence more secured against attacks. The correlation between the ciphered images for the original key and the ciphered images for the changed key for every stage is shown in Table II. To investigate the effectiveness of the decryption in the IFF block for incorrect fractional orders and , the MSE between the original images and the decrypted images for different values of and were obtained. Fig.4 shows the plot between fractional orders and the respective MSE values. It was seen that the MSE was very low (<100) for the correct orders of (=0.7) and (=0.4). However, for the incorrect values, the MSE between the original and the decrypted image was greater than 3.5x103. This suggests that the IFF algorithm is very

TABLE I. COMPARISON OF THE CORRELATION VALUES BETWEEN ADJACENT PIXELS FOR ORIGINAL AND ENCRYPTED IMAGES

STAGE Correlation between adjacent pixels Horizontal Vertical Diagonal

Stage 1 (Lena)

Original

Encrypted

0.9695

0.0008

0.9401

0.0059

0.9180

0.0029

Stage 2 (Bird)

Original

Encrypted

0.9812

-0.0018

0.9899

0.0023

0.9875

-0.0041

Stage 3

(Camera)

Original

Encrypted

0.9335

0.0014

0.9592

-0.0006

0.9293

0.0020

Fig.3. (a), (c), (e) represent the encrypted images obtained after stages 1, 2 and 3 of the algorithm and (b), (d), (f) represent their respective histograms. sensitive to the fractional orders ( and ) and the original image cannot be recovered efficiently with incorrect fractional orders. Also, a comparison between the decrypted images obtained using the correct phase key (MSE=27.6) and the slightly changed phase key (MSE= 6.7x103) with correct keys X1 and X2 is shown in Fig.5 which indicates that the image cannot be recovered effectively with a wrong phase key leading to the increased security provided by our algorithm. Table III shows a comparison of results obtained for the encrypted image of the proposed method with some existing methods.

TABLE II. COMPARISON OF THE CORRELATION BETWEEN ENCRYPTED IMAGES FOR DIFFERENT KEYS

STAGE

Correlation between encrypted images for different keys

MSB changed key and original

key

LSB changed key and original

key

MSB changed key and LSB changed key

Stage 1 (Lena)

Stage 2 (bird)

Stage 3

(camera)

0.0014

0.0056

0.0023

-0.0023

-0.0059

-0.0062

0.0005

-0.0011

0.0057

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.

Fig.4. (a) represents the plot between MSE (y-axis) and (x-axis), (b) represents the plot between MSE (y-axis) and (x-axis).

IV.CONCLUSION A new multi-stage multiple image encryption algorithm based on a combination of an Iterative Fractional Fourier Transform block and 2-round chaotic Block Cipher has been presented in this paper. Various security analysis such as the key space analysis, correlation analysis of adjacent pixels, histogram analysis and mean square error analysis were performed and our results showed that our proposed algorithm is highly efficient and resistant to attacks. A more secure encryption compared to previous multiple image encryption techniques was obtained and the errors in the decrypted images were found out to be very minimal (MSE<100). Moreover, without the correct keys and phase values at every stage, it was observed that the correct information could not be retrieved bolstering the fact that our system is highly secure. The main advantages of our proposed algorithm are that the efficiency of encryption does not decrease even if the number of images increases and that all the images (at different stages) are decrypted with the same quality.

TABLE III. COMPARISON WITH EXISTING METHODS

Parameter of Comparison

Proposed Algorithm

[3]

[11]

Multiple Image

Encryption

Yes

No

Yes

NPCR

0.9961

0.9869

0.9033

UACI

0.33562

0.3274

0.2810

Average Entropy

7.996

7.9854

7.0078

Cross

Correlation

-0.0021

0.0056

0.0093

Fig.5. (a) is the decrypted bird image (Stage 2) with the correct phase key (MSE =27.6) and (b) is the decrypted bird image with the slightly changed phase key (MSE = 6.7x103 ).

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