On Mignotte secret sharing schemes over Gaussian integers
DOI:
https://doi.org/10.13069/jacodesmath.v9i3.211Keywords:
Secret sharing, Mignotte sharing scheme, Gaussian integer, Chinese remainder theoremAbstract
Secret Sharing Schemes (SSS) are methods for distributing a secret among a set of participants. One of the first Secret Sharing Schemes was proposed by M. Mignotte, based on the Chinese remainder theorem over the ring of integers. In this article we extend the Mignotte's scheme to the ring of Gaussian Integers and study some of its properties. While doing this we aim to solve a gap in a previous construction of such extension. In addition we show that any access structure can be made through a SSS over $ \mathbb{Z}[i]$.
Received: 4 July 2021 | Accepted: 25 November 2021Downloads
References
C. A. Asmuth and J. Bloom, A modular approach to key safeguarding, IEEE Trans. Inform. Theory 29(2) (1983) 208–210.
R. E. Blahut, Algebraic Methods for Signal Processing and Communications Coding, Signal Processing and Digital Filtering, Springer-Verlag, New York, 1992.
C. Ding, D. Pei, and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography, World Scientific, Singapore, 1996.
J. B. Fraleigh, A First Course in Abstract Algebra, 2nd ed., Addison-Wesley, Reading, MA, 1976.
T. Galibus and G. Matveev, Generalized Mignotte’s sequences over polynomial rings, Electron. Notes Theor. Comput. Sci. 186 (2007) 43–48.
R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004.
S. Iftene, General secret sharing based on the Chinese remainder theorem with applications in e-voting, Electron. Notes Theor. Comput. Sci. 186 (2007) 67–84.
M. A. Jodeit, Jr., Uniqueness in the division algorithm, Amer. Math. Monthly 74(7) (1967) 835–836.
M. Mignotte, How to share a secret, in: T. Beth (Ed.), Cryptography, Lecture Notes in Computer Science, Vol. 149, Springer, Berlin, Heidelberg, 1983, pp. 371–375.
I. Ozbek, F. Temiz, and I. Siap, A generalization of the Mignotte’s scheme over Euclidean domains and applications to secret image sharing, J. Algebra Comb. Discrete Appl. 6(3) (2019) 147–161.
A. Shamir, How to share a secret, Commun. ACM 22(11) (1979) 612–613.
D. R. Stinson, Cryptography: Theory and Practice, 3rd ed., CRC Press, Boca Raton, FL, 2005.
G. Ulutas, M. Ulutas, and V. Nabiyev, Secret sharing scheme based on Mignotte’s scheme, in: 2011 IEEE 19th Signal Processing and Communications Applications Conference (SIU), IEEE, 2011, pp. 291–294.
