The Number Theoretic Hilbert Transform

This paper presents a general expression for a number theoretic Hilbert transform (NHT). The transformations preserve the circulant nature of the discrete Hilbert transform matrix together with alternating values in each row being zero and non-zero. Specific examples for 4-, 6-, and 8-point NHT are provided. The NHT transformation can be used as a primitive to create cryptographically useful scrambling transformations.

[1]  Subhash C. Kak,et al.  Multilayered array computing , 1988, Inf. Sci..

[2]  Rohith Singi Reddy.  Encryption of Binary and Non-Binary Data Using Chained Hadamard Transforms , 2010, ArXiv.

[3]  Trieu-Kien Truong,et al.  Convolutions over residue classes of quadratic integers , 1976, IEEE Trans. Inf. Theory.

[4]  C. Burrus,et al.  Number theoretic transforms to implement fast digital convolution , 1975, Proceedings of the IEEE.

[5]  S. Kak,et al.  On speech encryption using waveform scrambling , 1977, Bell Labs technical journal.

[6]  Subhash C. Kak,et al.  Multilevel Indexed Quasigroup Encryption for Data and Speech , 2009, IEEE Transactions on Broadcasting.

[7]  Charles H. Bennett,et al.  WITHDRAWN: Quantum cryptography: Public key distribution and coin tossing , 2011 .

[8]  S. Rajsbaum.  Foundations of Cryptography , 2014 .

[9]  Renuka Kandregula.  The Basic Discrete Hilbert Transform with an Information Hiding Application , 2009, ArXiv.

[10]  S. Kak.  The discrete Hilbert transform , 1970 .

[11]  Oded Goldreich.  Foundations of Cryptography: Index , 2001 .

[12]  Subhash C. Kak.  A Three-Stage Quantum Cryptography Protocol , 2005, ArXiv.

[13]  Subhash Kak,et al.  Hilbert transformation for discrete data , 1973 .

[14]  Pramode K. Verma,et al.  Multi-photon tolerant secure quantum communication — From theory to practice , 2013, 2013 IEEE International Conference on Communications (ICC).

[15]  Subhash C. Kak,et al.  A two-layered mesh array for matrix multiplication , 1988, Parallel Comput..

[16]  Oded Goldreich,et al.  Foundations of Cryptography: List of Figures , 2001 .

[17]  L. Scharf,et al.  Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals , 2010 .

[18]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[19]  Oded Goldreich,et al.  Foundations of Cryptography: Basic Tools , 2000 .

[20]  J. Pollard.  Implementation of number-theoretic transforms , 1976 .

[21]  S. K. Padala,et al.  Systolic arrays for the discrete Hilbert transform , 1997 .

[22]  Renuka Kandregula.  Towards a Number Theoretic Discrete Hilbert Transform , 2009, ArXiv.