Abstract
A black-box secret sharing scheme (BBSSS) for a given access structure works in exactly the same way over any finite Abelian group, as it only requires black-box access to group operations and to random group elements. In particular, there is no dependence on e.g. the structure of the group or its order. The expansion factor of a BBSSS is the length of a vector of shares (the number of group elements in it) divided by the number of players n.
At CRYPTO 2002 Cramer and Fehr proposed a threshold BBSSS with an asymptotically minimal expansion factor Θ(log n).
In this paper we propose a BBSSS that is based on a new paradigm, namely, primitive sets in algebraic number fields. This leads to a new BBSSS with an expansion factor that is absolutely minimal up to an additive term of at most 2, which is an improvement by a constant additive factor.
We provide good evidence that our scheme is considerably more efficient in terms of the computational resources it requires. Indeed, the number of group operations to be performed is Õ(n 2) instead of Õ(n 3) for sharing and Õ(n 1.6) instead of Õ(n 2.6) for reconstruction.
Finally, we show that our scheme, as well as that of Cramer and Fehr, has asymptotically optimal randomness efficiency.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Blakley, G.R.: Safeguarding cryptographic keys. In: Proc. National Computer Conference 1979. AFIPS Proceedings, vol. 48, pp. 313–317 (1979)
Cramer, R., Fehr, S.: Optimal black-box secret sharing over arbitrary Abelian groups. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 272–287. Springer, Heidelberg (2002)
Cramer, R., Fehr, S., Ishai, Y., Kushilevitz, E.: Efficient multi-party computation over rings. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 596–613. Springer, Heidelberg (2003)
Desmedt, Y., Frankel, Y.: Threshold cryptosystem. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 307–315. Springer, Heidelberg (1990)
Desmedt, Y., King, B., Kishimoto, W., Kurosawa, K.: A comment on the efficiency of secret sharing scheme over any finite abelian group. In: Boyd, C., Dawson, E. (eds.) ACISP 1998. LNCS, vol. 1438, pp. 391–402. Springer, Heidelberg (1998)
Fehr, S.: Secure Multi-Player Protocols: Fundamentals, Generality, and Efficiency. PhD thesis, University of Århus (2003)
Frankel, Y., Gemmell, P., MacKenzie, P., Yung, M.: Optimal resilience proactive public-key cryptosystems. In: Proceedings of FOCS 1997, pp. 384–393. IEEE Press, Los Alamitos (1997)
Karchmer, M., Wigderson, A.: On span programs. In: Proceedings of the Eigth Annual Structure in Complexity Theory Conference, pp. 102–111. IEEE Computer Society Press, Los Alamitos (1993)
King, B.: A Comment on Group Independent Threshold Sharing. In: Chae, K.-J., Yung, M. (eds.) WISA 2003. LNCS, vol. 2908, pp. 425–441. Springer, Heidelberg (2004)
King, B.S.: Some Results in Linear Secret Sharing. PhD thesis, University of Wisconsin-Milwaukee (2000)
Lang, S.: Algebra, 3rd edn. Addison-Wesley Publishing Company, Reading (1997)
Shamir, A.: How to share a secret. Communications of the ACM 22(11), 612–613 (1979)
Shoup, V.: Practical threshold signatures. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 207–220. Springer, Heidelberg (2000)
Stam, M.: Speeding up Subgroup Cryptosystems. PhD thesis, Technische Universiteit Eindhoven (2003)
Stinson, D., Wei, R.: Bibliography on Secret Sharing Schemes (2003), http://www.cacr.math.uwaterloo.ca/dstinson/ssbib.html
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cramer, R., Fehr, S., Stam, M. (2005). Black-Box Secret Sharing from Primitive Sets in Algebraic Number Fields. In: Shoup, V. (eds) Advances in Cryptology – CRYPTO 2005. CRYPTO 2005. Lecture Notes in Computer Science, vol 3621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11535218_21
Download citation
DOI: https://doi.org/10.1007/11535218_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28114-6
Online ISBN: 978-3-540-31870-5
eBook Packages: Computer ScienceComputer Science (R0)