Image Encryption Using Chaotic Cat Mapping in the Discrete Fourier Transform

—The paper presents an secure image using the two dimensional chaotic cat mapping (2D-CCM) in the Discrete Fourier Transform domain (DFT). The ciphering phase begins by applying theDFT on the plainimage to be encrypted and the resulted Fourier transformed image are scrambled using the 2D-CCM and finally an inverse DFT is applied to obtain the final encrypted image. The decryption phase applies a reverse procedure to get the original plainimage. A set of encryption test experiments are employed to inspect the proposed DFT based 2D-CCM image cryptosystem. The experimental results verified and confirmed the superiority of the proposed DFT based 2D-CCM image cryptosystem.


INTRODUCTION
Nowadays, due to the large utilization of digital images over telecommunication media, it is necessary to develop efficient and robust security techniques to secure them during transmission through communication systems [1][2][3][4]. The requirements to satisfy and fulfill the security requirements of digital images resulted in development of different encryption methods [4][5][6][7][8][9][10]. Within the last decade, several encryption methods have been suggested in the literature and they may be categorized into traditional and chaos based encryption methods. Traditional encryption methods may include DES, Double DES, Triple DES, IDEA, RC5, RC6 and [11][12][13][14]. But, these traditional methods does not allow a satisfactory outcomes due to of image intrinsic properties like large bulky capacity, high correlation, and redundancy [14][15][16].Chaos based encryption methods may be regarded as good practical tool as suchmethodspresent a good mix of high speed, and security, less complexity, allowable computational power and overheads [17][18].
The aim of this study is to introducing a secure DFT based 2D-CCM image cryptosystem that is efficiently capable of encrypting and decrypting digital images through secure or unsecure telecommunication networks. The encrypting phase starts through employing the DFT on the source plainimage and the obtained discrete Fourier transformed image is shuffled by a 2D-CCM and finally applying an inverse DFT to get the final ciphered image. The decrypting phase starts through employing the DFT on the encrypted image and the obtained discrete Fourier transformed image is inversely shuffled by an inverse 2D-CCM and finally applying an inverse DFT to get the final original image.
where mod, p , q are the modulo operation and control variables of the 2D-CCM which can be utilized as its secret keys.
After a set number of iterations m (modulo) of the 2D-CCM, the original image will return. The random relationship between the size of the image and the numbers of iterations which takes to return to the original image is depicted in Table 1 [21].

The DFT
The DFT may be regarded as an essential tool in digital signal processing. For two dimensional signal   y x z , of size M x N, the DFT and its inverse can be represented by [22]:

The Proposed DFT based 2D-CCM Image Cryptosystem
The

Security Study
This section is dedicated for testing the security of the proposed DFT based 2D-CCM image cryptosystem visually and with a group of encryption testing.The proposed DFT based 2D-CCM image cryptosystem is examined through a set of encryption tests to inspect the security of the proposed DFT based 2D-CCM image cryptosystem. These examination experiments have been performed with a group of testing image. The tested images may involve the Airplane, Boat and Cameraman images as illustrated in Fig. 4.

Histogram Testing
The histogram testing results of the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystem for the tested plainimages/cipherimages are illustrated in Fig. 6.The histogram testing results of the proposed DFT based 2D-CCM image cryptosystem for the encrypted images are almostdistinct from their respected original images histogram testing results. Also, it may be shown that thehistogram testing results of the 2D-CCM for encrypted images cipher are the same as the histogram testing results of their respected original images. This is expected since the 2D-CLM just changes the location of pixels.

Correlation Coefficients Testing
The correlation coefficients are computed between the encrypted and original images as [23]: Low correlation coefficients values ensure efficient encryption quality. The correlation coefficients testing results of the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystems for the tested plainimages/cipherimages are given in Table 1. The testing outcomes demonstrate that the correlation coefficients between pairs of plainimage/cipherimage are near the ideal zero value which in turn verified and ensured good encryption efficiency.

EntropyTesting
The entropy testing examines the information amount in the outcome encrypted image that resulted using the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystem The entropy may be computed using [24]: Where i x representsthe i th pixelgray value. High estimations for entropy demonstrate good encryption efficiency.
The entropy testing outcomes of the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystems are shown inTable 2. The entropy testing outcomes demonstrated that encrypted images entropy for the 2D-CCM are larger with respect to their respective results of the proposed DFT based 2D-CCM image cryptosystem. Table 2: Entropy outcomes of the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystems

Differential Testing
The Differential testing is conducted to examine the impact of employed for testing the effect of just onepixel alternation in the original image on the constructed encrypted image using cipherimage using the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystem. The differential testing involve the Numberof Pixels Change Rate (NPCR) and the Unified Average Changing Intensity (UACI) testing. The NPCR is estimated by [25][26][27]: (7) where N M , are both the height and width of Enc 1 and Enc 2 images.
The UACI is estimated by [25][26][27] (8) The NPCR and UACI testing outcomes of the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystems are shown in Table 3. The entropy testing outcomes demonstrated that encrypted images entropy for the 2D-CCM are larger with respect to their respective results of the proposed DFT based 2D-CCM image cryptosystem.
The NPCR and UACI testing outcomes ensured and provedthe high sensibility of both the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystems with respect very small alternations which also good encryption for both the 2D-CCM and the proposed DFT based 2D-CCM image cryptosystems.

Conclusion
The paper proposed a secure DFT based 2D-CCM image cryptosystem. The proposed DFT based 2D-CCM image cryptosystem is tested and examined and investigated with a group of different encrypting metrics like visual inspection, histogram, entropy, encryption quality and differential examinations.The experimental testsoutcomes confirmedthe efficiencyand the significance ofthe proposed DFT based 2D-CCM image cryptosystem.