Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Cryptography <s> The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Cryptography <s> Pei and Hsue (IEEE Signal Processing Letters, Vol. 13, No. 6, pp. 329-332, June 2006) proposed an encryption scheme based on multiple-parameter discrete fractional Fourier transform. We show that all the building blocks of this scheme are linear, and hence, breaking this scheme, using a known plaintext attack, is equivalent to solving a set of linear equations. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Cryptography <s> A novel image encryption algorithm is proposed based on the multiple-parameter fractional Fourier transform, which is a generalized fractional Fourier transform, without the use of phase keys. The image is encrypted simply by performing a multiple-parameter fractional Fourier transform with four keys. Optical implementation is suggested. The method has been compared with existing methods and shows superior robustness to blind decryption. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Cryptography <s> Methods of image encryption based on fractional Fourier transform have an incipient flaw in security. We show that the schemes have the deficiency that one group of encryption keys has many groups of keys to decrypt the encrypted image correctly for several reasons. In some schemes, many factors result in the deficiencies, such as the encryption scheme based on multiple-parameter fractional Fourier transform [Opt. Lett.33, 581 (2008)]. A modified method is proposed to avoid all the deficiencies. Security and reliability are greatly improved without increasing the complexity of the encryption process. © 2009 Optical Society of America <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Cryptography <s> In this letter, a new commuting matrix with random discrete Fourier transform (DFT) eigenvectors is first constructed. A random discrete fractional Fourier transform (RDFRFT) kernel matrix with random DFT eigenvectors and eigenvalues is then proposed. The RDFRFT has an important feature that the magnitude and phase of its transform output are both random. As an application example, a security-enhanced image encryption scheme based on the RDFRFT is illustrated. <s> BIB005
The FRFT can also be used in the field of cryptography  BIB001 BIB002 BIB003 BIB004 . For example, a 2004 US patent invented a cryptographic approach  . At the encryption side, a user first selects at least four parameters including an angle of rotation, a time exponent, a phase, and a sampling rate as the encryption key. Then, more than one modified FRFT kernels corresponding to the encryption key are selected and multiplied with the input signal. On the other hand, a reverse procedure with respect to the encryption process is used for decryption. The modified FRFT (e.g., BIB001 ) along with the double random phase encoding technique has been successfully applied for encrypting digital data. Specifically, using the double random phase encoding in the multiple-parameter FRFT domain, this encryption method enhances data security because the order parameters of the modified FRFT can be exploited as extra keys for decryption. However, this encryption scheme has shown to be linear BIB002 . Hence, it is insecure because a known plaintext attack can break this scheme, which is equivalent to solving a set of linear equations BIB002 . It is also shown that the current standard algorithms such as AES  outperform the above system in terms of both encryption speed, bandwidth, and storage requirements. Recently, a random discrete FRFT, proposed in BIB005 , exhibits an important feature that the magnitude and phase of its transform output are both random. This random discrete FRFT is also first applied to the image encryption to show its potentials.