Skip to main content
SpringerLink
Log in

Secret sharing schemes with partial broadcast channels

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In a conventional secret sharing scheme a dealer uses secure point-to-point channels to distribute the shares of a secret to a number of participants. At a later stage an authorised group of participants send their shares through secure point-to-point channels to a combiner who will reconstruct the secret. In this paper, we assume no point-to-point channel exists and communication is only through partial broadcast channels. A partial broadcast channel is a point-to-multipoint channel that enables a sender to send the same message simultaneously and privately to a fixed subset of receivers. We study secret sharing schemes with partial broadcast channels, called partial broadcast secret sharing schemes. We show that a necessary and sufficient condition for the partial broadcast channel allocation of a (t, n)-threshold partial secret sharing scheme is equivalent to a combinatorial object called a cover-free family. We use this property to construct a (t, n)-threshold partial broadcast secret sharing scheme with O(log n) partial broadcast channels. This is a significant reduction compared to n point-to-point channels required in a conventional secret sharing scheme. Next, we consider communication rate of a partial broadcast secret sharing scheme defined as the ratio of the secret size to the total size of messages sent by the dealer. We show that the communication rate of a partial broadcast secret sharing scheme can approach 1/O(log n) which is a significant increase over the corresponding value, 1/n, in the conventional secret sharing schemes. We derive a lower bound on the communication rate and show that for a (t,n)-threshold partial broadcast secret sharing scheme the rate is at least 1/t and then we propose constructions with high communication rates. We also present the case of partial broadcast secret sharing schemes for general access structures, discuss possible extensions of this work and propose a number of open problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N Alon (1986) ArticleTitleExplicit construction of exponential sized families of k-independent sets Disc Maths 58 191–193 Occurrence Handle0588.05003 Occurrence Handle829076 Occurrence Handle10.1016/0012-365X(86)90161-5

    Article  MATH  MathSciNet  Google Scholar 

  2. M Atici SS Magliveras DR Stinson WD Wei (1996) ArticleTitleSome recursive constructions for perfect hash families J Combin Designs 4 353–363 Occurrence Handle0914.68087 Occurrence Handle1402122 Occurrence Handle10.1002/(SICI)1520-6610(1996)4:5<353::AID-JCD4>3.0.CO;2-E

    Article  MATH  MathSciNet  Google Scholar 

  3. J Benaloh J Leichter (1988) ArticleTitleGeneralised secret sharing and monotone functions Adv Cryptol – CRYPTO ’ 88 IssueIDLNCS 403 27–35

    Google Scholar 

  4. Ben-Or M, Goldwasser S, Wigderson A (1988) Completeness theorems for non-cryptographic fault-tolerant distributed computing. Proc ACM STOC, ’88, ACM Press, pp 1–10

  5. Beimel A, Chor B (1998) Secret sharing with public reconstruction. IEEE Trans Info Theory 44(5):1887–1896 (Extended abstract in Crypto’95)

    Google Scholar 

  6. J Bierbrauer (1997) ArticleTitleUniversal hashing and geometric codes Designs Codes Cryptogr 11 207–221 Occurrence Handle0876.94044 Occurrence Handle1451727 Occurrence Handle10.1023/A:1008226810363

    Article  MATH  MathSciNet  Google Scholar 

  7. Blackburn SR (1999) Combinatorics and threshold cryptology, Combinatorial Designs and their Applications (Chapman and Hall/CRC Research Notes in Mathematics). CRC Press, London, pp 49–70

  8. SR Blackburn M Burmester Y Desmedt PR Wild (1996) ArticleTitleEfficient multiplicative sharing schemes Adv Cryptol – Eurocrypt ’96, LNCS 96 IssueID1070 107–118 Occurrence Handle1421582

    MathSciNet  Google Scholar 

  9. SR Blackburn PR Wild (1998) ArticleTitleOptimal linear perfect hash families J Combin Theory Ser A 83 233–250 Occurrence Handle0914.68088 Occurrence Handle1636981 Occurrence Handle10.1006/jcta.1998.2876

    Article  MATH  MathSciNet  Google Scholar 

  10. GR Blakey (1979) ArticleTitleSafeguarding cryptographic keys Proc AFIPS 1979 National Computer Conference 48 313–317

    Google Scholar 

  11. GR Blakely C Meadows (1984) ArticleTitleSecurity of ramp schemes Adv Cryptol: Crypto ’84, LNCS 196 242–268

    Google Scholar 

  12. C Blundo A Santi ParticleDe DR Stinson U Vaccaro (1995) ArticleTitleGraph decompositions and secret sharing schemes J Cryptol 8 39–64 Occurrence Handle0816.94013 Occurrence Handle10.1007/BF00204801

    Article  MATH  Google Scholar 

  13. Brickell EF (1991) A problem in broadcast encryption. Presented at 5th Vermont Summer Workshop on Combinatorics and Graph Theory, June 1991

  14. RM Capocelli A De Santis L Gargano U Vaccaro (1993) ArticleTitleOn the size of shares for secret sharing schemes J Cryptol 6 157–169 Occurrence Handle0786.68030 Occurrence Handle10.1007/BF00198463

    Article  MATH  Google Scholar 

  15. JL Carter MN Wegman (1979) ArticleTitleUniversal classes of hash functions JComp Syst Sci 18 143–154 Occurrence Handle0412.68090 Occurrence Handle532173 Occurrence Handle10.1016/0022-0000(79)90044-8

    Article  MATH  MathSciNet  Google Scholar 

  16. Chaum D, Crepeau C, Damgård I (1988) Multiparty unconditional secure protocols. Proc ACM STOC ’88, ACM Press, pp 11–19

  17. TM Cover JA Thomas (1991) Elements of Information Theory John Wiley & Sons New York Occurrence Handle0762.94001

    MATH  Google Scholar 

  18. ZJ Czech G Havas BS Majewski (1997) ArticleTitlePerfect hashing Theor Comp Sci 182 1–143 Occurrence Handle0954.68060 Occurrence Handle1463931 Occurrence Handle10.1016/S0304-3975(96)00146-6

    Article  MATH  MathSciNet  Google Scholar 

  19. Y Desmedt (1997) ArticleTitleSome recent research aspects of threshold cryptography 1997 Information Security Workshop, Japan (JSW ’97), LNCS 1396 99–114

    Google Scholar 

  20. Desmedt Y, Safavi-Naini R, Wang H, Chris C, Pieprzyk J (1999) Broadcast Anti-jamming systems, ICON ’99, IEEE International Conference on Networks. IEEE Computer Society, pp 349–355

  21. Desmedt Y, Jajodia S (1997) Redistributing secret shares to new access structures and its applications, Preprint

  22. P Erdös P Frankl Z Furedi (1985) ArticleTitleFamilies of finite sets in which no set is covered by the union of r others Israel J Math 51 79–89 Occurrence Handle0587.05021 Occurrence Handle804477

    MATH  MathSciNet  Google Scholar 

  23. ML Fredman J Komlòs (1984) ArticleTitleOn the size of separating systems and families of perfect hash functions SIAM J Alg Disc Methods 5 61–68 Occurrence Handle0525.68037

    MATH  Google Scholar 

  24. A Fiat M Naor (1994) ArticleTitleBroadcast encryption Adv Cryptol – Crypto ’93, LNCS 773 480–490 Occurrence Handle0870.94026

    MATH  Google Scholar 

  25. Franklin M, Yung M (1995) Secure hypergraphs: privacy from partial broadcast. Proc ACM STOC ’95, ACM Press, pp 36–44

  26. T Helleseth T Johansson (1996) ArticleTitleUniversal hash functions from exponential sums over finite fields and Galois Rings Adv Cryptol – Crypto’96, LNCS 96 IssueIDLNCS 1109 31–44 Occurrence Handle1480669

    MathSciNet  Google Scholar 

  27. Ito M, Saito A, Nishizeki T (1987) Secret sharing scheme realizing general access structure. Proceedings IEEE Global Telecommun. Conf., Globecom ’87, Washington, IEEE Communications Soc. Press, pp 99–102

  28. WA Jackson KM Martin (1996) ArticleTitleA combinatorial interpretation of ramp schemes Austral J Combin 14 51–60 Occurrence Handle0862.94016 Occurrence Handle1424321

    MATH  MathSciNet  Google Scholar 

  29. E Karnin J Greene M Hellman (1983) ArticleTitleOn secret sharing systems IEEE Trans Inform Theory 29 35–41 Occurrence Handle0503.94018 Occurrence Handle711276 Occurrence Handle10.1109/TIT.1983.1056621

    Article  MATH  MathSciNet  Google Scholar 

  30. R Kumar S Rajagopalan A Sahai (1999) ArticleTitleCoding constructions for blacklisting problems without computational assumptions Adv Cryptol – CRYPTO ’99, LNCS 1666 609–623 Occurrence Handle0942.94004

    MATH  Google Scholar 

  31. K Kurosawa T Yoshida (1999) ArticleTitleStrongly universal hashing and identification codes via channels IEEE Tran IT 45 IssueID6 2091–2095 Occurrence Handle0958.94012 Occurrence Handle1720661

    MATH  MathSciNet  Google Scholar 

  32. K Martin R Safavi-Naini H Wang (1999) ArticleTitleBounds and techniques for efficient redistribution of secret shares to new access structures Comp J 42 IssueID8 638–649 Occurrence Handle0955.68045 Occurrence Handle10.1093/comjnl/42.8.638

    Article  MATH  Google Scholar 

  33. Mehlhorn K (1984) Data structures and algorithms, vol 1. Springer-Verlag

  34. W. Ogata K Kurosawa (1998) ArticleTitleSome basic properties of general nonperfect secret sharing schemes J Universal Comp Sci 4 IssueID8 690–704 Occurrence Handle0967.68060 Occurrence Handle1654143

    MATH  MathSciNet  Google Scholar 

  35. M Ruszinko (1994) ArticleTitleOn the upper bound of the size of the r-cover-free families J Combin Theory A 66 302–310 Occurrence Handle0798.05071 Occurrence Handle1275736 Occurrence Handle10.1016/0097-3165(94)90067-1

    Article  MATH  MathSciNet  Google Scholar 

  36. A Shamir (1979) ArticleTitleHow to share a secret Commun ACM 22 612–613 Occurrence Handle0414.94021 Occurrence Handle549252 Occurrence Handle10.1145/359168.359176

    Article  MATH  MathSciNet  Google Scholar 

  37. DR Stinson (1992) ArticleTitleAn explication of secret sharing schemes Des Codes Cryptogr 2 357–390 Occurrence Handle0793.68111 Occurrence Handle1194776 Occurrence Handle10.1007/BF00125203

    Article  MATH  MathSciNet  Google Scholar 

  38. DR Stinson (1996) ArticleTitleOn the connection between universal hashing, combinatorial designs and error-correcting codes Congress Numerant 114 7–27 Occurrence Handle0897.68048 Occurrence Handle1421130

    MATH  MathSciNet  Google Scholar 

  39. DR Stinson T Trung Particlevan R Wei (2000) ArticleTitleSecure frameproof codes, key distribution patterns, group testing algorithms and related structures J Statist Plan Infer 86 595–617 Occurrence Handle1054.94013 Occurrence Handle10.1016/S0378-3758(99)00131-7

    Article  MATH  Google Scholar 

  40. DR Stinson R Wei L Zhu (2000) ArticleTitleNew constructions for perfect hash families and related structures using combinatorial designs and codes J Combin Designs 8 189–200 Occurrence Handle0956.68159 Occurrence Handle1752734 Occurrence Handle10.1002/(SICI)1520-6610(2000)8:3<189::AID-JCD4>3.0.CO;2-A

    Article  MATH  MathSciNet  Google Scholar 

  41. DR Stinson R Wei L Zhu (2000) ArticleTitleSome new bounds for cover-free families J Combin Theory A 90 224–234 Occurrence Handle0948.05055 Occurrence Handle1749434 Occurrence Handle10.1006/jcta.1999.3036

    Article  MATH  MathSciNet  Google Scholar 

  42. MN Wegman JL Carter (1981) ArticleTitleNew hash functions and their use in authentication and set equality J Comp Syst Sci 22 265–279 Occurrence Handle0461.68074 Occurrence Handle633535 Occurrence Handle10.1016/0022-0000(81)90033-7

    Article  MATH  MathSciNet  Google Scholar 

  43. C Xing H Wang KY Lam (2000) ArticleTitleConstructions of authentication codes from algebraic curves over finite fields IEEE Trans Info Theory 46 886–892 Occurrence Handle0997.94028 Occurrence Handle1763467 Occurrence Handle10.1109/18.841168

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Technology and Computer Science, University of Wollongong, Northfields Ave., Wollongong, NSW, 2522, Australia

    Rei Safavi-Naini

  2. Centre for Advanced Computing—Algorithms and Cryptography, Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Huaxiong Wang

  3. Division of Mathematical Sciences, Nanyang Technological University, Singapore

    Huaxiong Wang

Authors
  1. Rei Safavi-Naini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huaxiong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huaxiong Wang.

Additional information

Communicated by P. Wild.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Safavi-Naini, R., Wang, H. Secret sharing schemes with partial broadcast channels. Des Codes Crypt 41, 5–22 (2006). https://doi.org/10.1007/s10623-006-0027-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-006-0027-1

Keywords

AMS classifications

Search

Navigation