Abstract
Secret sharing has been a subject of study for over twenty years, and has had a number of real-world applications. There are several approaches to the construction of secret sharing schemes. One of them is based on coding theory. In principle, every linear code can be used to construct secret sharing schemes. But determining the access structure is very hard as this requires the complete characterisation of the minimal codewords of the underlying linear code, which is a difficult problem. In this paper we present a sufficient condition under which we are able to determine all the minimal codewords of certain linear codes. The condition is derived using exponential sums. We then construct some linear codes whose covering structure can be determined, and use them to construct secret sharing schemes with interesting access structures.
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References
R.J. Anderson, C. Ding, T. Helleseth, and T. Kløve, How to build robust shared control systems, Designs, Codes and Cryptography 15 (1998), pp.111–124.
A. Ashikhmin, A. Barg, G. Cohen, and L. Huguet, Variations on minimal codewords in linear codes, Proc.A AECC, 1995, pp. 96–105.
A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory 44(5) (1998), pp.2010–2017.
L.D. Baumert and W.H. Mills Uniform cyclotomy, Journal of Number Theory 14 (1982), pp.67–82.
L.D. Baumert and R.J. McEliece, Weights of irreducible cyclic codes, Information and Control 20(2) (1972), pp.158–175.
G.R. Blakley, Safeguarding cryptographic keys, Proc. NCC AFIPS, 1979, pp.313–317.
C. Ding and X. Wang, A coding theory construction of new Cartesian authentication codes, preprint, 2003.
R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997.
J.L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish-Russian Workshop on Information Theory, August 22–27, 1993, pp.276–279.
J.L. Massey, Some applications of coding theory in cryptography, Codes and Ciphers: Cryptography and Coding IV, Formara Ltd, Esses, England, 1995, pp.33–47.
R.J. McEliece and D.V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM 24 (1981), pp.583–584.
A. Renvall and C. Ding, The access structure of some secret-sharing schemes, Information Security and Privacy, Lecture Notes in Computer Science, vol.1172, pp.67–78, 1996, Springer-Verlag.
A. Shamir, How to share a secret, Comm. ACM 22 (1979), pp.612–613.
J. Yuan and C. Ding, Secret sharing schemes from two-weight codes, The Bose Centenary Symposium on Discrete Mathematics and Applications, Kolkata, Dec 2002.
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Ding, C., Yuan, J. (2003). Covering and Secret Sharing with Linear Codes. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds) Discrete Mathematics and Theoretical Computer Science. DMTCS 2003. Lecture Notes in Computer Science, vol 2731. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45066-1_2
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DOI: https://doi.org/10.1007/3-540-45066-1_2
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