Encryption of Images and Signals Using Wavelet Transform and Permutation Algorithm

Ibtisam A. Aljazaery

College Of Engineering/ Elect. Eng. Dep. Babylon University Hilla, Iraq

ABSTRACT:

In this paper, a new method to encrypt the signals with one dimension and images(monochrome or color images) in a time more less than if these signals and images are encrypted with their original sizes This method depends on extracting the important features which are distinguished these signals and images and then discarding them. The next step is encrypting  the lowest dimensions of these data. Discrete Wavelet transform (DWT) is used as a feature extraction because it is a powerful tool of signal processing for it’s multiresolutional possibilities. The chosen data is encrypted with one of conventional cryptographic algorithm (Permutation algorithm) after shrinking it’s dimension using suitable encryption key. The encrypted data is 100% unrecognized, besides, the decryption algorithm  returned  the encrypted data to it’s original dimension efficiently.

KEYWORDS: Cryptography; DWT; Permutation; Shrinking.

Copy the following to cite this article: Aljazaery I. A. Encryption of Images and Signals Using Wavelet Transform and Permutation Algorithm. Orient. J. Comp. Sci. and Technol;7(1)

Copy the following to cite this URL: Aljazaery I. A. Encryption of Images and Signals Using Wavelet Transform and Permutation Algorithm. Orient. J. Comp. Sci. and Technol;7(1). Available from: http://computerscijournal.org/?p=745

INTRODUCTION

Image encryption techniques are extensively used to overcome the problem of secure transmission for both images and text over the electronic media by using the conventional cryptographic algorithms. But the problem is that it cannot be used in case of huge amount of data and high resolution images[1,2]

Partial or selective encryption is the technique of securing the confidentiality of shrinking data by encrypting only a fraction of the total data to reduce computational overhead [3,9]. The most significant portion of the data, as dictated by a shrinking algorithm, is encrypted to disallow decoding without the knowledge of the decryption key.         

 The basic idea of this method is not to encrypt the whole signal or whole image, but, first: the signal with one dimension or two should analyzed using general wavelet transform formula, second: the image (monochrome or color) goes through the single level of (DWT) resulting four coefficients matrices, the approximation (ca),horizontal (ch),  vertical (cv) and diagonal (cd). Zeroing and shrinking algorithms are designed to get the encrypting results. To make the relevant signal or image more anonymously, a convention encrypted algorithm (Permutation algorithm) was used. This algorithm replaces the locations of signal’s and matrices’ elements. In decryption stage, the back-spacing algorithm was designed to return the earlier discarding values to their original locations of signals and images then, undergo the Inverse Discrete Wavelet Transform (IDWT) to produce unencrypted data. The main goal of this method is to reduce the encrypted time by only encrypting the insignificant part of signals and images.

DISCRETE WAVELET TRANSFORM (DWT)

Wavelets are mathematical functions that cut up data into different frequency components. Wavelet algorithms process data at different scales or resolutions. The wavelet transform carries out a special form of analysis by shifting the original signal from the time domain into the time–frequency, or, in this context, time–scale domain. It is illustrated in Figure(1). The idea behind the wavelet transform is the definition of a set of basis functions that allow an efficient, informative and useful representation of signals[4,10]. Discrete wavelet transform (DWT), which transforms a discrete time signal to a discrete wavelet representation. it converts an input series x₀, x₁, ..x_m, into one high-pass wavelet coefficient series and one low-pass wavelet coefficient series (of length n/2 each) given by [9, 10]:

formula1, 2

 

 

where s_m(Z) and t_m(Z) are called wavelet filters, K is the length of the filter, and i=0, …, [n/2]-1.

Lifting schema of (DWT) has been recognized as a faster approach. The basic principle is to factorize the polyphase matrix of a wavelet filter into a sequence of alternating upper and lower triangular matrices and a diagonal matrix, Figure (1). This leads to the wavelet implementation by means of banded-matrix multiplications, the related equations shown in (3) and (4)[5,6,11]. 

forumla3,4

 

 

fig1

 

 

 

where s_i(z) (primary lifting steps) and t_i(z) (dual lifting steps) are filters and (K) is a constant. As this factorization is not unique, several {si(z)}, {ti(z)} and (K) are admissible. The whole process is illustrated in Figure 2, (a) and (b)[5,6,12].

fig2

 

 

PERMUTATION CIPHER

A permutationcipher is a replacing of the locations of letters or numbers in the plaintext according to some specific system and key. A permutation, also called an “arrangement number” or “order,” is a rearrangement of the elements of an ordered list (S) into a one-to-one correspondence with (S) itself. The number of permutations on a set of (n) elements is given by (n)  factorial (n!)[7, 8].

PROPOSED SYSTEM

fig3

 

 

 

Figure 3(b) illustrates the proposed system when it works on images(mono. or color) using the single level of DWT. The lowest frequency sub-band is expressed in the matrix (ca) and the highest frequencies sub-bands are expressed in the matrices (ch), (cv) and (cd). Extract the highest entropy values from (c) in the first state or from (ca, ch, cv and cd) in the second state, and discarding them (using shrinking algorithm) is made  the signal or image lose its recognized features and it will be with low dimensions.

ALGORITHMS AND PSEUDO CODES OF THE SYSTEM

This section is about the practical side of this research, it illustrates all the relevant simulated algorithms “pseudo codes”, which explain how the system is working.

USING GENERAL FORMULA OF DWT WITH ONE DIMENSION SIGNALS

The following steps in algorithm (1) clarify zeroing the significant values of the signal and shrinking the rest of it after transform it from (time to frequency) domain using general formula of DWT.

table1

 

As below, algorithm (2) illustrates the Encryption and Decryption process using Permutation cipher algorithm

table2

 

 

After decrypt the shrinking signal, it must be back-space the original values to original location to obtain the expanded one dimensional signal (c), finally, this transformed signal (c) goes through IDWT to obtain the original signal (s). The above steps are explicated in back-spacing algorithm, as shown in algorithm(3) as below

table3

 

 

5.2 USING GENERAL FORMULA OF DWT WITH (IMAGES):

In this case, the main difference is using the general formula of DWT for two dimensional signals (images) instead of one dimensional formula, as in algorithm (1), which is [c,l]=wavedec2(X,3,’db1′), where (X) is a matrix representing the image, (3) is meaning three level decomposition, (db1) the wavelet name. So, algorithm(1), algorithm(2) and algorithm(3) are working here successfully. To retrieve the original signal(s), general formula of IDWT for two dimensional signals must use which is s=waverec2(c,l,’db1′), where (s) is a required signal, (c) is the vector of decomposition coefficients and (db1) is the same wavelet name used earlier.     

5.3 USING SINGLE LEVEL FORMULA OF DWT WITH IMAGES(MONO. AND COLOR):

Algorithm (4) elucidate the zeroing of the significant values of the four matrices (ca), (ch), (cv) and (cd) and shrinking the rest of them after transform monochrome image using single level formula of DWT as shown as below:

table4

 

 

 

The purpose of algorithm(5) is to generate the encrypting image as shown as below:

table5

 

 

 

The following algorithm contains, back-space the removed values to their original locations in (ca), (ch), (cv) and (cd) and gathering them using (IDWT) to get the decrypting monochrome image, algorithm (6) clarifies the above steps as below:

table6

 

 

 

For color image (i.e. three dimensional array), algorithm(4), algorithm(5) and algorithm(6) could be written again with some changes listed below:

table7

 

 

 

table8

 

 

 

table9

 

 

 

RESULTS AND DISCUSSION

The experimental results of this research will be declared in this section according to the algorithms which mentioned earlier in the previous section.

The result for the first case, (section 5.1), illustrates in the following figures (from 4(a) to 4(h)):

fig4a,b,c,d

 

 

 

fige,f,g,h

 

 

 

fig5

 

 

 

fig5

 

 

 

 

fig6

 

 

 

fig7

 

 

 

 

table10

 

 

 

From table (1) and figures from (9(a to c)), we can notice the mean values of the original and encrypted signals and images are different,and it always has less value of encrypted signals and images, because of the shrinking of original data and discarding the highest values from them. Besides, it’s value in decrypted signal or image is like to original value,  that is lead us to conclude notation mentioned before which is the proposed method was able to return the encrypted image to it’s original dimensions and values.

fig8a,b,c,d

 

 

 

table11

 

 

table12

 

 

 

The aim of this work is security, that means confidentiality and robustness against attacks to break the images or signals.  Less data to encrypt means less CPU time required forencryption. So, the proposed system used the proposed technique which resulted in reducing encryption and decryption time. The signal was used has 1500 samples   instead of 2500 samples of the original signal and the dimensions of the monochrome image  are (156 x 156)  instead of (256 x 256) of the original woman image, finally, color image has (264 x 264 x 3) dimensions instead of (328 x 450 x 3). of Iraq image.  A good image or signal encryption algorithm should besensitive to the cipher key and it has to be large enough tomake brute-force attack infeasible. In current work, the size of cipher key of the first case (one dimensional signal ) is (1 x 750)and it’s size of the others is (1 x 833).

fig9

 

 

 

CONSLUSIONS & FUTURE WORK

The following observations are concluded from the above graphs and tables:

1. Figures from (8(a) to 8(d)), explain the histograms of the used images with their decryption,   obviously the amplitude of the decrypted images less than the originals, even the color distributing is different between them.
2. From table(2)  and figure 9(a),we can notice that the entropy values for original signal and images are more than the entropy values of encrypted signal and images, that is because discarding of the high values(less information for the encrypted signals and images), and shrinking them to another dimensions.
3. Woman image is used in two cases in this research, the entropy value of encrypted woman image of the first case is equal to zero while it’s value of the second case is 0.1303 table(2), this simple comparison forces us to conclude the first case is more efficient in encryption than the second case.
4. Spending time in encryption and decryption, that is what table (3) tell us about. Time of encryption is more than time of decryption because, encryption pass through more than step using more than algorithm ,mentioned earlier, to get accepted encrypted results. Although the decryption algorithms which are used from encrypted user are not easy to get the original signals and images, but they are implemented in simple way with less steps using one of key management methods to get the hoped original signal and image.      

About future work, some of flashed ideas could be used, embedded the basic idea of this research to get may be, enhanced or developed results, as follows:

1. Using the essential idea with videos.
2. Using another transformers like Principle Component Analysis PCA.

Using another conventional or unconventional cryptographic algorithms. 

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