Steerable Fourier number transform with application to image encryption

Abstract In this paper, we introduce the steerable Fourier number transform (SFNT). It corresponds to a generalization of the well-known number-theoretic transform (NTT), obtained by applying a finite field rotation to pairs of vectors taken from the original Fourier-like NTT basis. We provide illustrative examples regarding the SFNT construction and propose an image encryption scheme based on the new transform; the principle of the method is to use the angles of the referred rotations as secret parameters. We demonstrate that the proposed scheme is resistant against the main cryptographic attacks, although it is simpler than other transform-based image encryption techniques.

[1]  Wen-Liang Hsue,et al.  Real Discrete Fractional Fourier, Hartley, Generalized Fourier and Generalized Hartley Transforms With Many Parameters , 2015, IEEE Transactions on Circuits and Systems I: Regular Papers.

[2]  Vassil S. Dimitrov,et al.  Fragile watermarking using finite field trigonometrical transforms , 2009, Signal Process. Image Commun..

[3]  Yicong Zhou,et al.  Medical image encryption using high-speed scrambling and pixel adaptive diffusion , 2018, Signal Process..

[4]  Sergio Callegari,et al.  Fundamental performance limits of chaotic-map random number generators , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[5]  R. Blahut.  Fast Algorithms for Signal Processing , 2010 .

[6]  Juliano B. Lima,et al.  Finite field trigonometric transforms , 2011, Applicable Algebra in Engineering, Communication and Computing.

[7]  Mahmoud H. Annaby,et al.  Image encryption via discrete fractional Fourier-type transforms generated by random matrices , 2016, Signal Process. Image Commun..

[8]  Jing Ma,et al.  Security of image encryption scheme based on multi-parameter fractional Fourier transform , 2016 .

[9]  Kwok-Wo Wong,et al.  Period Distribution of the Generalized Discrete Arnold Cat Map for $N = 2^{e}$ , 2013, IEEE Transactions on Information Theory.

[10]  Yu-Guang Yang,et al.  Novel Image Encryption based on Quantum Walks , 2015, Scientific Reports.

[11]  J. B. Lima.  Fast algorithm for computing cosine number transform , 2015 .

[12]  Pierre Vandergheynst,et al.  Graph Signal Processing: Overview, Challenges, and Applications , 2017, Proceedings of the IEEE.

[13]  L. Kocarev,et al.  Chaos-based random number generators. Part II: practical realization , 2001 .

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

[15]  Vijay Kumar,et al.  Colour image encryption technique using differential evolution in non-subsampled contourlet transform domain , 2018, IET Image Process..

[16]  Enrico Magli,et al.  Steerable Discrete Fourier Transform , 2017, IEEE Signal Processing Letters.

[17]  Yao Tianren,et al.  Algorithm for linear convolution using number theoretic transforms , 1988 .

[18]  David Burton.  Elementary Number Theory , 1976 .

[19]  S. Agaian,et al.  NPCR and UACI Randomness Tests for Image Encryption , 2011 .

[20]  Ran Tao,et al.  Multichannel Random Discrete Fractional Fourier Transform , 2015, IEEE Signal Processing Letters.

[21]  Hejiao Huang,et al.  2D Logistic-Sine-coupling map for image encryption , 2018, Signal Process..

[22]  P. Duhamel,et al.  REALITY PRESERVING FRACTIONAL TRANSFORMS , 2004 .

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

[24]  Yasmine Abouelseoud,et al.  Two-Phase Image Encryption Scheme Based on FFCT and Fractals , 2017, Secur. Commun. Networks.

[25]  L. Kocarev,et al.  Chaos-based random number generators-part I: analysis [cryptography] , 2001 .

[26]  J. Pollard,et al.  The fast Fourier transform in a finite field , 1971 .

[27]  Juliano B. Lima,et al.  Image encryption based on the fractional Fourier transform over finite fields , 2014, Signal Process..

[28]  Yicong Zhou,et al.  Design of image cipher using block-based scrambling and image filtering , 2017, Inf. Sci..

[29]  Wei Zhang,et al.  Colour image encryption algorithm using one-time key and FrFT , 2018, IET Image Process..

[30]  Enrico Magli,et al.  Steerable Discrete Cosine Transform , 2017, IEEE Trans. Image Process..

[31]  Anlong Ming,et al.  Reality-Preserving Multiple Parameter Discrete Fractional Angular Transform and Its Application to Color Image Encryption , 2019, IEEE Transactions on Circuits and Systems for Video Technology.

[32]  Trieu-Kien Truong,et al.  The use of finite fields to compute convolutions , 1975, IEEE Trans. Inf. Theory.

[33]  Xuejing Kang,et al.  Color Image Encryption Using Pixel Scrambling Operator and Reality-Preserving MPFRHT , 2019, IEEE Transactions on Circuits and Systems for Video Technology.

[34]  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.

[35]  Xiaojun Tong,et al.  Hyperchaotic system-based pseudorandom number generator , 2016, IET Inf. Secur..

[36]  Amr M. Youssef.  On the Security of a Cryptosystem Based on Multiple-Parameters Discrete Fractional Fourier Transform , 2008, IEEE Signal Processing Letters.

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

[38]  R. Amutha,et al.  Cosine Number Transform based hybrid image compression-encryption , 2016, 2016 International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET).