The multiple-parameter discrete fractional Fourier transform

Abstract

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 order parameters, where 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.

Hybrid image encryption based on digital pre-encryption and optical single random phase encoding

“…The sensitivity of the encryption key to the fractional order parameters of the DFRFT is tested by making a small error δ ranging from -0.04 to +0.04 in the two parametric orders a' = a + δ and b' = b + δ and we calculate the corresponding MSE. based on DFRFT-SRPE firms the superiority of the proposed method when it is compared with the method of the reference [12], and with the DRPE method based on FRFT [9].…”

“…Since its first appearance by REFRIGIER and JAVIDI in 1995 [1], based on the bidirectional Fourier transform (FT), the DRPE has expanded to several modifications. Parametric transforms [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] have been introduced instead of bidirectional FT and their independent parameters are beneficially exploited as an additional secret key, amongst them, the reciprocal-orthogonal parametric (ROP) transform [2][3][4], the fractional Fourier transform (FRFT) [5][6][7][8], the multiple discrete fractional Fourier transform (MDFRFT) [9][10][11][12], gyrator transform [13], Fresnel transform [14], discrete parametric Fourier transform [15] and angular transform [16].…”

Optical multiple-image encryption in discrete multiple-parameter fractional Fourier transform scheme using complex encoding, theta modulation and spectral fusion

“…for N even and D denotes a diagonal matrix whose diagonal entries correspond to the eigenvalues for each column of eigenvectors v k in V [27][28][29], and T denotes the matrix transpose. In our encryption system, the encryption is performed digitally or optically, while the decryption can be processed digitally in a computer.…”

Physical Layer Authenticated Image Encryption for IoT Network Based on Biometric Chaotic Signature for MPFrFT OFDM System

“…One key feature of IoT systems is their ability to support a variety of legacy and emerging communication protocols, including SigFox, cellular technology, 6LoWPAN (IPv6 Low-power Wireless Personal Area Networks (LoWPAN)), BLE (Bluetooth low energy), ZigBee, RFID (radio frequency identification), NFC (near-field communication), Z-Wave, NB-IoT (Narrow Band IoT), LoRaWAN (long-range wide area network), and Wi-SUN (wireless smart utility network) [41]. There are currently eight major categories of PLS schemes that concentrate on data confidentiality for OFDM systems: channel-based encryption [42], phase encryption [43], permutation [44,45], artificial noise (AN) and artificial fast fading (AFF) [46,47], preamble modulation [48] (Figure 1), power allocation [49], Peak-to-Average Power Reduction (PAPR) encryption [50], the frequency domain [51] and the time domain [52] are two other areas in which these techniques can be used. Chaos-based physical layer encryption is used in OFDM-based IoT systems to achieve the phase randomization and constellation rotation in the transmitted image in both spatial and transformation domains.…”