A generalization of the Mignotte’s scheme over Euclidean domains and applications to secret image sharing

Year 2019, , 147 - 161, 13.09.2019

İbrahim Ozbek Fatih Temiz İrfan Siap

https://doi.org/10.13069/jacodesmath.617239

Abstract

Secret sharing scheme is an efficient method to hide secret key or secret image by partitioning it into parts such that some predetermined subsets of partitions can recover the secret but remaining subsets cannot. In 1979, the pioneer construction on this area was given by Shamir and Blakley independently. After these initial studies, Asmuth-Bloom and Mignotte have proposed a different $(k,n)$ threshold modular secret sharing scheme by using the Chinese remainder theorem. In this study, we explore the generalization of Mignotte's scheme to Euclidean domains for which we obtain some promising results. Next, we propose new algorithms to construct threshold secret image sharing schemes by using Mignotte's scheme over polynomial rings. Finally, we compare our proposed scheme to the existing ones and we show that this new method is more efficient and it has higher security.

Keywords

Mignotte sequences, Secret image sharing, Secret sharing scheme, Euclidean domain

References

• [1] C. Asmuth, J. Bloom, A modular approach to key safeguarding, IEEE Trans. Inform. Theory 29(2) (1983) 208–210.
• [2] G. R. Blakley, Safeguarding cryptographic keys, Proc. Am. Federation of Information Processing Soc. (AFIPS’79) National Computer Conf. 48 (1979) 313–317.
• [3] P. Dingyi, S. Arto, D. Cunsheng, Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography, World Scientific, 1996.
• [4] T. Hungerford, Abstract Algebra: An Introduction, Cengage Learning, Boston, 2012.
• [5] S. Iftene, General secret sharing based on the Chinese Remainder Theorem with applications in E–Voting, Electronic Notes in Theoretical Computer Science 186 (2007) 67–84.
• [6] E. V. Krishnamurthy, Error–Free Polynomial Matrix Computations, Springer Science and Business Media, New York, 2012.
• [7] J. B. Lima, R. M. Campello de Souza, Histogram uniformization for digital image encryption, 25th SIBGRAPI Conference on Graphics, Patterns and Images (2012) 55–62.
• [8] P. K. Meher, J. C. Patra, A new approach to secure distributed storage, sharing and dissemination of digital image, IEEE International Symposium on Circuits and Systems (2006) 373-376.
• [9] M. Mignotte, How to share a secret,In: Cryptography, EUROCRYPT 1982, Lecture Notes in Computer Science 149 (1983) 371–375.
• [10] M. Naor, A. Shamir, Visual cryptography, In: Advances in Cryptology–EUROCRYPT 1994, Lecture Notes in Computer Science 950 (1994) 1–12.
• [11] O. Ore, The general Chinese remainder theorem, The American Mathematical Monthly 59(6) (1952) 365–370.
• [12] A. Shamir, How to share a secret, Comm. ACM 22(11) (1979) 612–613.
• [13] S. J. Shyu, Y. R. Chen, Threshold secret image sharing by Chinese remainder theorem,IEEE Asia– Pacific Services Computing Conference (2008) 1332–1337.
• [14] D. R. Stinson, An explication of secret sharing schemes, Des. Codes Cryptogr. 2(4) (1992) 357–390.
• [15] S. Somaraj, M. A. Hussain, Performance and Security Analysis for Image Encryption using Key Image, Indian Journal of Science and Technology 8(35) (2015).
• [16] G. Tatyana, M. Genadii, Generalized Mignotte’s sequences over polynomial rings, Electronic Notes in Theoretical Computer Science 186 (2007) 43–48.
• [17] C. C. Thien, J. C. Lin, Secret image sharing, Comput. Graph. 26(5) (2002) 765–770.
• [18] G. Ulutas, M. Ulutas, V. Nabiyev, Secret sharing scheme based on Mignotte’s scheme, 2011 IEEE 19th Signal Processing and Communications Applications Conference (2011) 291–294.
• [19] R. Z. Wang, C. H. Su, Secret image sharing with smaller shadow images,Pattern Recognition Lett. 27(6) (2006) 551–555.
• [20] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing 13(4) (2004) 600–612.