Fractional Fourier, Hartley, Cosine and Sine Number-Theoretic Transforms Based on Matrix Functions

In this paper, we introduce fractional number-theoretic transforms (FrNTT) based on matrix functions. In contrast to previously proposed FrNTT, our approach does not require the construction of any number-theoretic transform (NTT) eigenvectors set. This allows us to obtain an FrNTT matrix by means o...

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Bibliographic Details
Published in:Circuits, systems, and signal processing systems, and signal processing, 2017-07, Vol.36 (7), p.2893-2916
Main Authors: Lima, Paulo Hugo E. S., Lima, Juliano B., Campello de Souza, Ricardo M.
Format: Article
Language:English
Subjects:
Circuits and Systems
Eigenvectors
Electrical Engineering
Electronics and Microelectronics
Encryption
Engineering
Instrumentation
Mathematical analysis
Matrix methods
Signal,Image and Speech Processing
Transformations (mathematics)
Trigonometric functions
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Summary:In this paper, we introduce fractional number-theoretic transforms (FrNTT) based on matrix functions. In contrast to previously proposed FrNTT, our approach does not require the construction of any number-theoretic transform (NTT) eigenvectors set. This allows us to obtain an FrNTT matrix by means of a closed-form expression corresponding to a linear combination of integer powers of the respective NTT matrix. Fractional Fourier, Hartley, cosine and sine number-theoretic transforms are developed. We show that fast algorithms applicable to ordinary NTT can also be used to compute the proposed FrNTT. Furthermore, we investigate the relationship between fractional Fourier and Hartley number-theoretic transforms, and demonstrate the applicability of the proposed FrNTT to a recently introduced image encryption scheme.
ISSN:0278-081X
1531-5878
DOI:10.1007/s00034-016-0447-8