A Lossless Linear Algebraic Secret Image Sharing Scheme

Kanso, Ali; Ghebleh, Mohammad; Alazemi, Abdullah

doi:10.15388/20-INFOR420

Informatica

A Lossless Linear Algebraic Secret Image Sharing Scheme

Volume 31, Issue 3 (2020), pp. 499–522

Ali Kanso Mohammad Ghebleh Abdullah Alazemi

https://doi.org/10.15388/20-INFOR420

Pub. online: 17 June 2020 Type: Research Article

Open Access

Received
1 September 2019

Accepted
1 May 2020

Published
17 June 2020

Abstract

A $(k,n)$-threshold secret image sharing scheme is any method of distributing a secret image amongst n participants in such a way that any k participants are able to use their shares collectively to reconstruct the secret image, while fewer than k shares do not reveal any information about the secret image. In this work, we propose a lossless linear algebraic $(k,n)$-threshold secret image sharing scheme. The scheme associates a vector ${\mathbf{v}_{i}}$ to the ith participant in the vector space ${\mathbb{F}_{{2^{\alpha }}}^{k}}$, where the vectors ${\mathbf{v}_{i}}$ satisfy some admissibility conditions. The ith share is simply a linear combination of the vectors ${\mathbf{v}_{i}}$ with coefficients from the secret image. Simulation results demonstrate the effectiveness and robustness of the proposed scheme compared to standard statistical attacks on secret image sharing schemes. Furthermore, the proposed scheme has a high level of security, error-resilient capability, and the size of each share is $1/k$ the size of the secret image. In comparison with existing work, the scheme is shown to be very competitive.

References

Arnol’d, V.I., Avez, A. (1968). Ergodic Problems of Classical Mechanics. W. A. Benjamin Inc., New York, Amsterdam. Translated from the French by A. Avez.

Asmuth, C., Bloom, J. (1983). A modular approach to key safeguarding. IEEE Transactions on Information Theory, 29(2), 208–210.

Bai, L. (2006). A reliable (k, n) image secret sharing scheme. In: 2006 2nd IEEE International Symposium on Dependable, Autonomic and Secure Computing. IEEE, pp. 31–36.

Bassham, L. III., Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E., Leigh, S., Levenson, M., Vangel, M., Banks, D., Heckert, N., Dray, J. (2010). Sp 800-22 Rev. 1a. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications.

Blakley, G.R. et al. (1979). Safeguarding cryptographic keys. In: Proceedings of the National Computer Conference, Vol. 48.

Chang, C.C., Hwang, R.J. (1998). Sharing secret images using shadow codebooks. Information Sciences, 111(1–4), 335–345.

Chang, C.C., Chan, C.S., Fan, Y.H. (2006). A secret image sharing scheme based on vector quantization mechanism. In: International Conference on Embedded and Ubiquitous Computing. Springer, pp. 469–478.

Chang, C.C., Hsieh, Y.P., Lin, C.H. (2008). Sharing secrets in stego images with authentication. Pattern Recognition, 41(10), 3130–3137.

Chen, G., Mao, Y., Chui, C.K. (2004). A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos, Solitons & Fractals, 21(3), 749–761.

Chen, L.S.T., Su, W.K., Lin, J.C. (2009). Secret image sharing based on vector quantization. International Journal of Circuits, Systems and Signal Processing, 3(3), 137–144.

del Rey, A.M. (2008). A matrix-based secret sharing scheme for images. In: Iberoamerican Congress on Pattern Recognition. Springer, pp. 635–642.

Deng, X., Wen, W., Shi, Z. (2017). Threshold multi-secret sharing scheme based on phase-shifting interferometry. Optics Communications, 387, 409–414.

Desmedt, Y., Frankel, Y. (1991). Shared generation of authenticators and signatures. In: Annual International Cryptology Conference. Springer, pp. 457–469.

Ding, W., Liu, K., Yan, X., Wang, H., Liu, L., Gong, Q. (2018). An image secret sharing method based on matrix theory. Symmetry, 10(10), 530.

El-Latif, A.A.A., Yan, X., Li, L., Wang, N., Peng, J.L., Niu, X. (2013). A new meaningful secret sharing scheme based on random grids, error diffusion and chaotic encryption. Optics & Laser Technology, 54, 389–400.

Elsheh, E., Hamza, A.B. (2010). Comments on matrix-based secret sharing scheme for images. In: Iberoamerican Congress on Pattern Recognition. Springer, pp. 169–175.

Fridrich, J., Goljan, M., Du, R. (2002). Lossless data embedding—new paradigm in digital watermarking. EURASIP Journal on Advances in Signal Processing, 2002(2), 986842.

Fu, C., Zhang, G.Y., Zhu, M., Chen, J.X., Lei, W.M. (2018). A fast chaos-based colour image encryption algorithm using a hash function. Informatica, 29(4), 651–673.

Gersho, A., Gray, R.M. (2012). Vector Quantization and Signal Compression, Vol. 159. Springer Science & Business Media.

Ghebleh, M., Kanso, A. (2014). A robust chaotic algorithm for digital image steganography. Communications in Nonlinear Science and Numerical Simulation, 19(6), 1898–1907.

Ghebleh, M., Kanso, A. (2018). A novel secret image sharing scheme using large primes. Multimedia Tools and Applications, 77(10), 11903–11923.

Gray, R. (1984). Vector quantization. IEEE ASSP Magazine, 1(2), 4–29.

Hu, W.T., Li, M.C., Guo, C., Ren, Y.Z. (2012). Reversible secret image sharing with steganography and dynamic embedding. Security and Communication Networks, 5(11), 1267–1276.

Kabirirad, S., Eslami, Z. (2018). A (t, n)-multi secret image sharing scheme based on Boolean operations. Journal of Visual Communication and Image Representation, 57, 39–47.

Kanso, A., Ghebleh, M. (2012). A novel image encryption algorithm based on a 3D chaotic map. Communications in Nonlinear Science and Numerical Simulation, 17(7), 2943–2959.

Kanso, A., Ghebleh, M. (2013). A fast and efficient chaos-based keyed hash function. Communications in Nonlinear Science and Numerical Simulation, 18(1), 109–123.

Kanso, A., Ghebleh, M. (2015). A structure-based chaotic hashing scheme. Nonlinear Dynamics, 81(1–2), 27–40.

Kanso, A., Ghebleh, M. (2017). An efficient (t, n)-threshold secret image sharing scheme. Multimedia Tools and Applications, 76(15), 16369–16388.

Kanso, A., Ghebleh, M. (2018). An efficient lossless secret sharing scheme for medical images. Journal of Visual Communication and Image Representation, 56, 245–255.

Le, T.H.N., Lin, C.C., Chang, C.C., Le, H.B. (2011). A high quality and small shadow size visual secret sharing scheme based on hybrid strategy for grayscale images. Digital Signal Processing, 21(6), 734–745.

Lin, C.C., Tsai, W.H. (2004). Secret image sharing with steganography and authentication. Journal of Systems and Software, 73(3), 405–414.

Maneri, C., Silverman, R. (1966). A vector-space packing problem. Journal of Algebra, 4(3), 321–330.

Mignotte, M. (1982). How to share a secret. In: Workshop on Cryptography. Springer, pp. 371–375.

Naor, M., Wool, A. (1998). Access control and signatures via quorum secret sharing. IEEE Transactions on Parallel and Distributed Systems, 9(9), 909–922.

Rong, C.G., Xiaoning, D. (1998). From Chaos to Order: Methodologies, Perspectives and Applications, Vol. 24. World Scientific.

Schinzel, A., Spież, S., Urbanowicz, J. (2010). Admissible tracks in Shamir’s scheme. Finite Fields and Their Applications, 16(6), 449–462.

Schoenmakers, B. (1999). A simple publicly verifiable secret sharing scheme and its application to electronic voting. In: Annual International Cryptology Conference. Springer, pp. 148–164.

Shamir, A. (1979). How to share a secret. Communications of the ACM, 22(11), 612–613.

Shannon, C.E. (1951). Prediction and entropy of printed English. Bell System Technical Journal, 30(1), 50–64.

Simić, N., Perić, Z.H., Savić, M.S. (2018). Image coding algorithm based on Hadamard transform and simple vector quantization. Multimedia Tools and Applications, 77(5), 6033–6049.

Spiez, S., Srebrny, M., Urbanowicz, J. (2009). Secret sharing matrices. Preprint http://www.impan.pl/Preprints/p708.pdf.

Spież, S., Srebrny, M., Urbanowicz, J. (2012). Remarks on the classical threshold secret sharing schemes. Fundamenta Informaticae, 114(3–4), 345–357.

Thien, C.C., Lin, J.C. (2002). Secret image sharing. Computers & Graphics, 26(5), 765–770.

Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P. et al. (2004). Image quality assessment: from error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4), 600–612.

Wu, K.S. (2013). A secret image sharing scheme for light images. EURASIP Journal on Advances in Signal Processing, 2013(1), 49.

Wu, Y., Noonan, J.P., Agaian, S. et al. (2011). NPCR and UACI randomness tests for image encryption Cyber Journals: Multidisciplinary Journals in Science and Technology, Journal of Selected Areas in Telecommunications (JSAT), 1(2), 31–38.

Wylie, J.J., Bigrigg, M.W., Strunk, J.D., Ganger, G.R., Kiliccote, H., Khosla, P.K. (2000). Survivable information storage systems. Computer, 33(8), 61–68.

Zarepour-Ahmadabadi, J., Ahmadabadi, M.S., Latif, A. (2016). An adaptive secret image sharing with a new bitwise steganographic property. Information Sciences, 369, 467–480.

Biographies

Kanso Ali

ali.kanso@ku.edu.kw

A. Kanso is an associate professor of mathematics at Kuwait University, Kuwait. He received his BSc degree in mathematics from Queen Mary and Westfield College (University of London), in 1994. He earned his MSc degree in applied computing technology at the Electronic Engineering department of Middlesex University, in 1996. In 1999 he obtained a PhD in mathematics from Royal Holloway and Bedford New College (University of London). His research interests include chaos-based encryption systems, information hiding, hash functions, secret sharing, and graph theory.

Ghebleh Mohammad

mohammad.ghebleh@ku.edu.kw

M. Ghebleh is an associate professor of mathematics at Kuwait University, Kuwait. He received his BSc and MSc in mathematics from Sharif University of Technology, Tehran, Iran (1997 and 1999), and his PhD in mathematics from Simon Fraser University, Burnaby, British Columbia, Canada (2007). His research interests include graph theory, combinatorics, and digital security topics such as encryption, data hiding, hash functions, and secret sharing.

Alazemi Abdullah

abdullah.alazemi@ku.edu.kw

A. Alazemi is an associate professor of mathematics at Kuwait University, Kuwait. He received his BSc in mathematics from Kuwait University, Kuwait. He earned his MSc and PhD in mathematics from Colorado State University, Colorado, the United States (2004 and 2007). His research interests include incidence structures, classification problems, spectral graph theory, graph theory, combinatorics and algebra.

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Keywords

secret sharing secret image sharing (k, n)-threshold scheme admissible tracks chaos

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