Data ecryption based on multi-order FrFT, and FPGA implementation of DES algorith

Received Apr 16, 2019 Revised Apr 25, 2019 Accepted Apr 04, 2020 Cryptography techniques need some algorithms for encryption of data. Most of available encryption techniques are used for textual data; a few of encryption methods are used for multimedia data; However, This Algorithms that are used for textual data may not be inefficient for multimedia, because it is size is greater than the text. Therefore, Cryptosystems need to find and develop a new encryption schemes for such data. The most popular symmetric key algorithms are Data Encryption Standard (DES). However, DES is may be not suitable for multimedia because it consumes times. Encryption and decryption of these data require different methods. In this paper a method for encryption/decryption data by using the nature of FrFT in signals analysis, based on multi-order Fractional Fourier Transform has been introduced. The security of the method used in the encryption work was taken into account to identify the different indicators to measure the security of the encryption Techniques. These indicators are: sensitivity proposed Techniques for the key, the complexity of the processes, and statistical analysis. The key is formed by combination of order of Fractional Fourier Transform. The encrypted data is obtained by the summation of different orders. Numerical simulation results are given to demonstrate this proposed method.


INTRODUCTION
The science of protecting the information by converting it into unreadable while stored and transmitted is Cryptography [1]. The encryption is plays a major role in securing the data in transmission. Different encryption techniques are used to protect confidential data from unauthorized uses. Cryptography technique needs some algorithms for encryption of data [2]. One of the most popular symmetric key algorithms are Data Encryption Standard (DES).
A 64-bits key are used with DES, while 128,192,256 bits keys uses for AES [3]. DES, AES offer the greatest security to sensitive data compared to other cryptographic algorithms. The AES was accepted as a standard in November 2001 [4].
One of the most popular tools used in signal processing and analysis are The Fourier transform (FT) [5]. The idea of fractional powers of the Fourier operator appears in the mathematical literature as early as 1929 [6][7][8]. It has been rediscovered in quantum mechanics [9,10], optics [11][12][13], and signal processing

Theory of fractional fourier transform
The Fourier transform is a rotation by angle /2 in the time-frequency plane, the fractional Fourier transform interpreted as the counterclockwise rotation by an angle α in the time-frequency plane. FRFT is the generalization of the classical FT. Conventionally, FrFT of α order of input function x(t) can be defined as follows [17]: x(t) signal recovered by FrFT operation with backward angels -α: And, by substituting y for x and v for u, y-axis, The signal f(x,y) can be recovered by FrFT operation with backward angles(-αx ,-αy):

PROPOSED APPROACH
The proposed encryption technique is shown in Figure 3. Let original data S represents the input data to be encrypted Using FrFT. In Encryption steps based on FrFT, we use one -dimensional analysis to describe our methods, then we can extend all formulae to Two-dimensions. To obtain encrypted data, firstly, input data is multiplied by matrix R, and their results are transformed through first FrFT system with first order of transform a1 to get data, the result from this stage is transformed through second order of transform a2 by taking second FrFT, then it passes the result by taking FrFT with third order of transform a3 to get encrypted data 'L', The encrypted data is obtain by summations of different orders, and the key for encryption/decryption process is a combination of order of Fractional Fourier Transform [18] and matrix R. Encryption model as shown in Figure 3. is secure and more robust towards brute force attack, but the complexity of the system is increased.
In decryption process, the reverse for encryption process, is applied as shown in Figure 4. Firstly, transformed encrypted data 'L' through first FrFT with order of transform -a3, and passes the result again through second FrFT with order of transform -a2 , then the result from last stage passes through last FrFT with order of transform -a1 , finally, the result is multiplied with matrix conjugate of the matrix R* to get input data S.

Example 1: Enc/Dec proposed method for an image:
Mathematically, the encryption process in Figure 4, Summarized as: input data S (x, y) is multiplied with matrix R and passes through FrFT with order a1 till ak; as in equations bellow: ( , ) ' xy L S R =  xy L F F S R aa = (11) Encrypted data is given by: The decryption process is in figure 14 and mathematically is given as: Finally; decrypted image is given by: The time in seconds for encryption and decryption operations of an Image, and audio signals are shown in Table 1 and Table 2.

Example 2: Enc/Dec proposed method for audio files
Another type of data is an audio signal. Audio cryptography encryption is the method of including the key to the plain audio, while decryption is the process of taking out the original plain back by using the same key. In this part we offer the possibility of encryption and decryption for data by using our proposed methods with FrFT for an audio signal, through use of the nature of FrFT in signals analysis, Figue 5 and Figure 6. shows original audio1 and histogram. Figure 7. is shows encrypted audio1 signal with FrFT, while Figure 8. shows decrypted audio1 signal (reconstructed audio1).

COMPARATIVE STUDY BETWEEN DES & FrFT USING SOFTWARE SIMULATIOM
Computer simulations have been done of the proposed encryption technique, the image present in study is shown in Figure 9., is "Image1" and 140 x 200 pixels, and 28.8 KB size of data; and the matrix R; orders of transform a1, a2, And a3. This sub section, the security analysis of the proposed method, and statistical analysis have been introduced. A simulation result has been discussed and it will be seen that data encryption based on FrFT provided criteria for security.

Security analysis 4.1.1. Key space analysis:
Let us suppose that an encryption scheme has k-bit key. So, an attacker needs 2 k operations to determine the key. The key in our proposed encryption methods based on FrFT is formed by combination of orders of transform values of Fractional Fourier Transform and matrix R.

Sensitivity to key:
If we encrypt data by K1 and decrypt by a different key K2, decryption should be unsuccessful. Our Proposed method is sensitive to the key changes. Figure 10. and Figure 11. shows decryption with correct and incorrect keys.  Figure 11. Decrypted image1 with incorrect key

Information hiding:
One of the important properties provided by encryption techniques is information hiding. It means that no information of original data can be extracted from the encrypted. Our encryption method is encrypting data successfully, and no information of original data can be extracted as shown in Figure 12 and Figure 13.

Statistical analysis:
Statistical analysis has been performed by calculating the histograms.

Histograms analysis:
one of the important features in data statistical analysis is Histogram. When encrypted data have a uniform histogram distribution, no useful information according to the statistical properties can be obtained. Figure 14 and Figure 15 are two gray images: Image1 and Image4, respectively. The histograms in Figure 16 and Figure 17 illustrate the original images pixels distributed at each gray level. The histogram of two images is very different as shown in Figure 16 and Figure 17. The histograms of the encrypted images Image1 and Image4 as shown in Figure 18 and Figure 19 are quite similar.
To evaluate an encryption scheme and encryption quality Peak signal-to noise ratio (PSNR) can be used. PSNR is usually expressed in decibels. Mathematically, PSNR can be described in (18) The experimental results that show the MSE and PSNR for an Image are shown in Table 2, Table 3, and Table 4 shows the peak signal to noise ratio (db). The experimental results that show the MSE and PSNR for an Audio are shown in Table 5 and Table 6.

Differential analysis
In image encryption, the cipher resistance to differential attacks is commonly analyzed via the NPCR and UACI tests [19]. number of pixels change rate while one pixel of plain image is changed is refers to Number of Pixels Change Rate (NPCR). To determines the average intensity of differences between the plain and ciphered, Unified Average Changing Intensity (UACI) is using. The NPCR and the UACI are defined in (19) and (21): where C1 and C2 are two cipher-images whose plain are different by only one bit. The gray value at grid (i; j) in C1 and C2, by C1(i; j) and C2(i; j). D array is determined by C1(i; j) and C2(i; j), if C1(i; j) = C2(i; j) then D(i; j) = 0 ; otherwise, D(i; j) = 1. D (i, j) is defined as in (20)

Correlation coefficient analysis
The relationship and similarity between two variables are described by Correlation. If correlation coefficient is equal to one, then two data are identical and they are in perfect correlation, In case of perfect correlation (correlation coefficient is equal to 1). Encryption process completely fails because the encrypted data is same as the plain data, When correlation coefficient is -1 then encrypted is negative of original (plain). Mathematically correlation coefficient can be shown as in (22) The values of the Correlation Coefficient between original, encrypted, and decrypted Images are showing in Table 9 and Table 10. The values of the Correlation Coefficient between original, encrypted, and decrypted audio are showing in Table 11 and Table 12.   Table 13 shows a comparative study of an Image, between DES, AES [22], and proposed encryption-decryption system based-on FrFT.

Xilinx platform studio
FPGA implemented is a simple design using VHDL and Verilog on FPGA. The applications will be written using "C or C++", finally to run the applications on processor system download bit file to the FPGA.

Simulation results of DES algorithm:
In this section, analysis of the simulation results has been introduced, Figure 20 and Figure 21. shows Xilinx Platform Studio and Graphical design view. Figure 22. shows the block of DES algorithms showing input and output pins, in this we give 64-bit data and 64-bit key, so that it gives us 64-bit encrypted data after whole encryption 16 rounds. Figure 23. is show the simulated result of encryption and decryption on ISE13.1.

CONCLUSIONS
Nowadays the security of data in the digital world becomes more and more important. When using the encryption algorithms may add burdens on the system: in the design process, the financial cost, or in processing time according to the complexity of the internal processes that make up the algorithm, and this is to be expected.
A method to encrypt and decrypt data based on multi-order Fractional Fourier Transform has been introduced. For more security, and to make encryption model robust towards brute force attack the key is formed by combination of order of Fractional Fourier Transform and the matrix, data has been encrypted and decrypted data successfully. The Key space analysis, statistical analysis, and key sensitivity analysis have been carrying out to demonstrate the security of the data encryption techniques. To evaluate the encryption performance and quality of the proposed schemes, the mean square error (MSE), PSNR, NPCR and UACI have been introduced. to measure the number of changing pixels and the number of averaged changed intensity between cipher data, the NPCR and UACI have been introduced, respectively. Implementation of DES algorithm using FPGA has been introduced. FPGA Implantation of DES Algorithms, and the simulation result, was performed on Xilinx ISE for implementing encryption and decryption module, with NEYXS 3 FPGA board.
We do not compare the globally agreed between systems and between the use of encryption (FrFT) to encrypt data. We do not say that the use of FrFT for encrypt data is better, but trying to offer a safe way without adding additional costs or burdens and up nearly applicable safety standard in encryption systems. The future work will be: implementation of FrFT encryption and decryption model using FPGA.