Image encryption based on double random amplitude coding in random Hartley transform domain
Introduction
Optical image encryption is an important tool in information security. Many algorithms based on Fourier transform or fractional Fourier transforms (FrFTs) had been proposed [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. The commonly adopted algorithms employed pure random phase masks, such as double random phase encoding [3], [4], to add noise information into the image. One of the drawbacks of such image encryption scheme is that the encrypted images are complex and hence inconvenient in real applications. Recently, an algorithm was proposed by Chen and Zhao [17] that can obtain real encrypted data with Hartley transform. They have introduced a pure random intensity mask as a filter at Hartley transform plane in optics. However, the security of the algorithm is weak against the bare attack, a decryption method without key [18]. In order to withstand the attack of bare decryption, the random intensity mask should be properly chosen [19] in the algorithm [17], which is not convenient for the application of image encryption.
More recently we have proposed a new random transform deduced from the FT, the random Fourier transform (RFT), by assigning a random phase directly to the eigenvalues of the FT [20]. Such a randomization of the eigenvalues in FT can be clearly seen from the ambiguity of the eigenvalues for FrFTs when the fractional order becomes an irrational. Based on the RFT, random Hartley transform (RHT) is, therefore, defined and presented in this paper. An image encryption scheme is given by using RHTs together with double random amplitude filtering. The new algorithm can realize an encryption code in real number domain and it is more robust against bare decryption attack.
The remainder of the paper is organized in the following sequence: In Section 2 we present the definition of RHT and discuss the encryption algorithm in detail. Section 3 is concerned with the numerical simulation results of image encryption. Conclusions are drawn in Section 4.
Section snippets
Image encryption algorithm
The RFT of 1-D function can be defined as [20]where is the decomposition coefficient based on Hermite–Gaussian polynomials, i.e. . , which are random numbers distributed on the unit circle on the complex plane, are the eigenvalues of the RFT. is the -th order normalized Hermite–Gaussian function. As we know, the Hartley transform of a real function can be defined as
Numerical results
Numerical simulation studies are carried out to evaluate the performance of the proposed scheme and discrete RHT is calculated by using discrete RFT [20]. The elements of the random matrices are taken from the range of that satisfies the uniform distribution. A photograph of Elaine with the size is adopted as the original image to be encrypted as shown in Fig. 4(a) for simulation. Fig. 4(b) gives a picture of encrypted image by using RHT with different sets of eigenvalues.
The security of the proposed algorithm
In general case, attacker has only an image, namely the encrypted image but two random matrices and RHTs are unknown. Situ et al. [21] had studied the security of double random phase encoding algorithm. However, the random amplitude encoding is different from the random phase encoding in theory. For instance, from a complex function , which denotes the product of the original image and phase function, attacker can easier separate encrypted image and phase by taking module as following
Conclusion
In this paper we have considered a concept of random Hartley transform (RHT) based on the definition of RFT and an image encryption algorithm based on this concept has been proposed. A significant feature of this algorithm is that the encrypted image can be expressed as a real number and hence can be conveniently stored and transferred. A double random amplitude coding scheme in RHT domain has also been introduced to be employed along with the proposed algorithm. It has been shown that the
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant nos. 10674038 and 10604042 and National Basic Research Program of China under Grant 2006CB302901, and China Postdoctoral Science Foundation (20080430913) development program for outstanding young teachers in Harbin Institute of Technology (HITQNJS.2008.027).
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