Abstract
A secret sharing scheme (SSS) was introduced by Shamir in 1979 using polynomial interpolation. Later it turned out that it is equivalent to an SSS based on a Reed–Solomon code. SSSs based on linear codes have been studied by many researchers. However there is little research on SSSs based on additive codes. In this paper, we study SSSs based on additive codes over GF(4) and show that they require at least two steps of calculations to reveal the secret. We also define minimal access structures of SSSs from additive codes over GF(4) and describe SSSs using some interesting additive codes over GF(4) which contain generalized 2-designs.
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References
Assmus, E.F., Mattson, H.F.: New 5-designs. J. Comb. Theory 6, 122–151 (1969)
Blakley, G.R.: Safeguarding cryptographic keys. In: National Computer Conference, pp. 313–317. American Federation of Information Processing Societies (1979)
Carreras, F., Magaña, A., Munuera, C.: The accessibility of an access structure. RAIRO Theor. Inf. Appl. 40(04), 559–567 (2006)
Delsarte, P.: Four fundamental parameters of a code and their combinatorial significance. Inf. Control 23, 407–438 (1973)
Ding, C., Kohel, D.R., Ling, S.: Secret-sharing with a class of ternary codes. Theor. Comput. Sci. 246(1), 285–298 (2000)
Ding, C., Yuan, J.: Covering and secret sharing with linear codes. In: Calude C. S., Dinneen M. J., Vajnovszki V (eds.) Discrete Mathematics and Theoretical Computer Science, pp. 11–25. Springer, Heidelberg (2003)
Dougherty, S.T., Mesnager, S., Solé, P.: Secret-sharing schemes based on self-dual codes. In: Information Theory Workshop. ITW’08. IEEE (2008)
Höhn, G.: Self-dual codes over the Kleinian four group. Math. Ann. 327(2), 227–255 (2003)
Huffman, W. C., Gaborit, P., Kim, J.-L., Pless, V.: On additive \(GF (4)\) codes. In: Codes and Association Schemes: DIMACS Workshop Codes and Association Schemes, November 9–12, 1999, DIMACS Center. vol. 56. American Mathematical Society (2001)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes, pp. 291–337. Cambridge University Press, New York (2003)
Kim, J.-L., Pless, V.: Designs in additive codes over \(GF(4)\). Des. Codes Cryptogr. 30(2), 187–199 (2003)
Li, Z., Xue, T.X., Lai, H.: Secret sharing schemes from binary linear codes. Inf. Sci. 180(22), 4412–4419 (2010)
MacWilliams, F.J., Odlyzko, A.M., Sloane, N.J.A., Ward, H.N.: Self-dual codes over \(GF(4)\). J. Comb. Theory Ser. A 25(3), 288–318 (1978)
Massey, J. L.: Minimal codewords and secret sharing. In: Proceedings 6th Joint Swedish–Russian International Workshop on Information Theory, pp. 276–279 (1993)
McEliece, R.J., Sarwate, D.V.: On sharing secrets and Reed–Solomon codes. Commun. ACM 24(9), 583–584 (1981)
Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)
Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)
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J.-L. Kim was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005172). N. Lee was partially supported by the National Institute for Mathematical Sciences (No. A21503).
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Kim, JL., Lee, N. Secret sharing schemes based on additive codes over GF(4). AAECC 28, 79–97 (2017). https://doi.org/10.1007/s00200-016-0296-5
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DOI: https://doi.org/10.1007/s00200-016-0296-5