Advertisement

A New (k,n)-Threshold Secret Sharing Scheme and Its Extension

  • Jun Kurihara
  • Shinsaku Kiyomoto
  • Kazuhide Fukushima
  • Toshiaki Tanaka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5222)

Abstract

In Shamir’s (k,n)-threshold secret sharing scheme (threshold scheme), a heavy computational cost is required to make n shares and recover the secret. As a solution to this problem, several fast threshold schemes have been proposed. This paper proposes a new (k,n)-threshold scheme. For the purpose to realize high performance, the proposed scheme uses just EXCLUSIVE-OR(XOR) operations to make shares and recover the secret. We prove that the proposed scheme is a perfect secret sharing scheme, every combination of k or more participants can recover the secret, but every group of less than k participants cannot obtain any information about the secret. Moreover, we show that the proposed scheme is an ideal secret sharing scheme similar to Shamir’s scheme, which is a perfect scheme such that every bit-size of shares equals that of the secret. We also evaluate the efficiency of the scheme, and show that our scheme realizes operations that are much faster than Shamir’s. Furthermore, from the aspect of both computational cost and storage usage, we also introduce how to extend the proposed scheme to a new (k,L,n)-threshold ramp scheme similar to the existing ramp scheme based on Shamir’s scheme.

Keywords

Secret sharing scheme threshold scheme threshold ramp scheme exclusive-or entropy random number ideal secret sharing scheme 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proc. AFIPS, vol. 48, pp. 313–317 (1979)Google Scholar
  3. 3.
    Blakley, G.R., Meadows, C.: Security of ramp schemes. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 242–269. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  4. 4.
    Yamamoto, H.: On secret sharing systems using (k,L,n) threshold scheme. IEICE Trans. Fundamentals (Japanese Edition) J68-A(9), 945–952 (1985)Google Scholar
  5. 5.
    Kurosawa, K., Okada, K., Sakano, K., Ogata, W., Tsujii, T.: Non perfect secret sharing schemes and matroids. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 126–141. Springer, Heidelberg (1994)Google Scholar
  6. 6.
    Ogata, W., Kurosawa, K.: Some basic properties of general nonperfect secret sharing schemes. J. Universal Computer Science 4(8), 690–704 (1998)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Okada, K., Kurosawa, K.: Lower bound on the size of shares of nonperfect secret sharing schemes. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 34–41. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  8. 8.
    Ishizu, H., Ogihara, T.: A study on long-term storage of electronic data. In: Proc. IEICE General Conf., vol. D-9-10(1), p. 125 (2004) (in Japanese)Google Scholar
  9. 9.
    Fujii, Y., Tada, M., Hosaka, N., Tochikubo, K., Kato, T.: A fast (2,n)-threshold scheme and its application. In: Proc. CSS 2005, pp. 631–636 (2005) (in Japanese)Google Scholar
  10. 10.
    Hosaka, N., Tochikubo, K., Fujii, Y., Tada, M., Kato, T.: (2,n)-threshold secret sharing systems based on binary matrices. In: Proc. SCIS. pp. 2D1–4 (2007) (in Japanese)Google Scholar
  11. 11.
    Kurihara, J., Kiyomoto, S., Fukushima, K., Tanaka, T.: A fast (3,n)-threshold secret sharing scheme using exclusive-or operations. IEICE Trans. Fundamentals, E91-A(1), 127–138 (2008)CrossRefGoogle Scholar
  12. 12.
    Shiina, N., Okamoto, T., Okamoto, E.: How to convert 1-out-of-n proof into k-out-of-n proof. In: Proc. SCIS 2004, pp. 1435–1440 (2004) (in Japanese)Google Scholar
  13. 13.
    Kunii, H., Tada, M.: A note on information rate for fast threshold schemes. In: Proc. CSS 2006, pp. 101–106 (2006) (in Japanese)Google Scholar
  14. 14.
    Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Trans. Inform. Theory 29(1), 35–41 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Capocelli, R.M., De Santis, A., Gargano, L., Vaccaro, U.: On the size of shares for secret sharing schemes. J. Cryptology 6, 35–41 (1983)Google Scholar
  16. 16.
    Blundo, C., De Santis, A., Gargano, L., Vaccaro, U.: On the information rate of secret sharing schemes. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 149–169. Springer, Heidelberg (1993)Google Scholar
  17. 17.
    Stinson, D.R.: Decomposition constructions for secret sharing schemes. IEEE Trans. Inform. Theory 40(1), 118–125 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Stinson, D.R.: Cryptography: Theory and Practice. CRC Press, Florida (1995)zbMATHGoogle Scholar
  19. 19.
    Poettering, B.: SSSS: Shamir’s Secret Sharing Scheme, http://point-at-infinity.org/ssss/

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jun Kurihara
    • 1
  • Shinsaku Kiyomoto
    • 1
  • Kazuhide Fukushima
    • 1
  • Toshiaki Tanaka
    • 1
  1. 1.KDDI R&D Laboratories, Inc.SaitamaJapan

Personalised recommendations