Generalized Secret Sharing Schemes Using N\(^\mu \)MDS Codes

  • Sanyam Mehta
  • Vishal SaraswatEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989)


Mehta et al. [11] recently proposed an \({{\,\mathrm{NMDS}\,}}\) code-based secret sharing scheme having a richer access structure than the traditional (tn) threshold secret sharing schemes, and is based on two mutually nonmonotonic sets of user groups of sizes t and \(t-1\) respectively, where \(n \ge t > 1\) corresponds to the total number of users. We give a full generalization of their scheme with complete security proofs. We propose an efficient generalized secret sharing scheme constructed using \({{\,\mathrm{N^{\mu }MDS}\,}}\) codes with time complexity of \(O(n^3)\). The scheme accepts an access structure constructed using \(\mu +1\) mutually nonmonotonic sets of user groups with sizes, \(t, t-1, \dots , t-\mu \), respectively, where \(1 \le \mu < t\), and the parameter t defines the threshold such that all user groups of size greater than t can recover the secret. The proposed secret sharing scheme is perfect and ideal and has robust cheating detection and cheater identification features.


Secret sharing schemes Generalized access structure Near MDS codes Almost MDS codes 



The authors acknowledge the support of the Department of Mathematics, BITS Goa, Indian Institute of Technology, Jammu, and R. C. Bose Centre for Cryptology and Security, ISI Kolkata.


  1. 1.
    Benaloh, J., Leichter, J.: Generalized secret sharing and monotone functions. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 27–35. Springer, New York (1990). Scholar
  2. 2.
    Blakley, G.: Safeguarding cryptographic keys. In: AFIPS, pp. 313–317. AFIPS Press (1979)Google Scholar
  3. 3.
    Blakley, G., Kabatiansky, G.: Generalized ideal secret-sharing schemes and matroids. Probl. Peredachi Informatsii 33(3), 102–110 (1997)MathSciNetGoogle Scholar
  4. 4.
    Dijk, M.: A linear construction of perfect secret sharing schemes. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 23–34. Springer, Heidelberg (1995). Scholar
  5. 5.
    Dodunekov, S.: Applications of near MDS codes in cryptography. In: Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, NATO Science for Peace and Security Series - D: Information and Communication Security, vol. 23, pp. 81–86. IOS Press (2009)Google Scholar
  6. 6.
    Harn, L., Lin, C.: Detection and identification of cheaters in \((t, n)\) secret sharing scheme. Des. Codes Cryptograph. 52(1), 15–24 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  8. 8.
    Ito, M., Saito, A., Nishizeki, T.: Secret sharing scheme realizing general access structure. Electron. Commun. Jpn. (Part III: Fundam. Electron. Sci.) 72(9), 56–64 (1989)Google Scholar
  9. 9.
    Massey, J.: Minimal codewords and secret sharing. In: Sixth Joint Swedish-Russian Workshop on Information Theory, Molle, Sweden, pp. 276–279 (1993)Google Scholar
  10. 10.
    McEliece, R., Sarwate, D.: On sharing secrets and Reed-Solomon codes. Commun. ACM 24(9), 583–584 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mehta, S., Saraswat, V., Sen, S.: Secret sharing using near-MDS codes. In: Carlet, C., Guilley, S., Nitaj, A., Souidi, E.M. (eds.) C2SI 2019. LNCS, vol. 11445, pp. 195–214. Springer, Cham (2019). Scholar
  12. 12.
    Pieprzyk, J., Zhang, X.-M.: Ideal threshold schemes from MDS codes. In: Lee, P.J., Lim, C.H. (eds.) ICISC 2002. LNCS, vol. 2587, pp. 253–263. Springer, Heidelberg (2003). Scholar
  13. 13.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Viswanath, G., Rajan, B.S.: Matrix characterization of generalized Hamming weights. In: IEEE International Symposium on Information Theory, p. 61. IEEE (2001)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Goldman Sachs Services Pvt LtdBangaloreIndia
  2. 2.Robert Bosch Engineering & Business Solutions Pvt LtdBangaloreIndia

Personalised recommendations