A generating matrix method for constructing Hermite-Gaussian-like number-theoretic transform eigenvectors

Abstract In this paper, we present a method based on generating matrices for constructing number-theoretic transform (NTT) eigenvectors. We provide a specific generating matrix and, using it, we demonstrate how to construct an orthogonal basis of Hermite–Gaussian-like NTT eigenvectors, which can be used to define a fractional number-theoretic transform (FrNTT). We discuss the main aspects of the referred eigenbasis, present an example and suggest a potential application for the corresponding FrNTT.

[1]  Soo-Chang Pei,et al.  Generating matrix of discrete Fourier transform eigenvectors , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[2]  Soo-Chang Pei,et al.  Closed-Form Orthogonal DFT Eigenvectors Generated by Complete Generalized Legendre Sequence , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[3]  C. Burrus,et al.  Fast Convolution using fermat number transforms with applications to digital filtering , 1974 .

[4]  Juliano B. Lima,et al.  The fractional Fourier transform over finite fields , 2012, Signal Process..

[5]  Juliano B. Lima,et al.  Encryption of medical images based on the cosine number transform , 2015, Signal Process. Image Commun..

[6]  Richard E. Blahut,et al.  Fast Algorithms for Signal Processing: Bibliography , 2010 .

[7]  Alexey Kuznetsov.  Explicit Hermite-type Eigenvectors of the Discrete Fourier Transform , 2015, SIAM J. Matrix Anal. Appl..

[8]  Juliano B. Lima,et al.  Closed-form Hermite-Gaussian-like number-theoretic transform eigenvectors , 2016, Signal Process..

[9]  Ilsun You,et al.  A Novel NTT-Based Authentication Scheme for 10-GHz Quantum Key Distribution Systems , 2016, IEEE Transactions on Industrial Electronics.

[10]  Juliano B. Lima,et al.  Image encryption based on the finite field cosine transform , 2013, Signal Process. Image Commun..

[11]  Jun Cheng,et al.  Finite Field Spreading for Multiple-Access Channel , 2014, IEEE Transactions on Communications.

[12]  Janne Heikkilä,et al.  Video filtering with Fermat number theoretic transforms using residue number system , 2006, IEEE Transactions on Circuits and Systems for Video Technology.

[13]  D. Birtwistle.  The eigenstructure of the number theoretic transforms , 1982 .

[14]  Soo-Chang Pei,et al.  Closed-Form Orthogonal Number Theoretic Transform Eigenvectors and the Fast Fractional NTT , 2011, IEEE Transactions on Signal Processing.

[15]  J. McClellan,et al.  Eigenvalue and eigenvector decomposition of the discrete Fourier transform , 1972 .

[16]  Eugene I. Bovbel,et al.  The modified number theoretic transform over the direct sum of finite fields to compute the linear convolution , 1998, IEEE Trans. Signal Process..

[17]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 2000, IEEE Trans. Signal Process..

[18]  D. Panario,et al.  The eigenstructure of finite field trigonometric transforms , 2011 .

[19]  Yue Wang,et al.  Sparse Discrete Fractional Fourier Transform and Its Applications , 2014, IEEE Transactions on Signal Processing.

[20]  Juliano B. Lima,et al.  Discrete Fractional Fourier Transforms Based on Closed-Form Hermite–Gaussian-Like DFT Eigenvectors , 2017, IEEE Transactions on Signal Processing.

[21]  Juliano B. Lima,et al.  Fractional Fourier, Hartley, Cosine and Sine Number-Theoretic Transforms Based on Matrix Functions , 2017, Circuits Syst. Signal Process..

[22]  Soo-Chang Pei,et al.  Optimal Discrete Gaussian Function: The Closed-Form Functions Satisfying Tao’s and Donoho’s Uncertainty Principle With Nyquist Bandwidth , 2016, IEEE Transactions on Signal Processing.

[23]  Fernando Pérez-González,et al.  Number Theoretic Transforms for Secure Signal Processing , 2016, IEEE Transactions on Information Forensics and Security.

[24]  Dorota Majorkowska-Mech,et al.  A Low-Complexity Approach to Computation of the Discrete Fractional Fourier Transform , 2017, Circuits Syst. Signal Process..