Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 623–644 | Cite as

Optimal assignment schemes for general access structures based on linear programming

  • Qiang LiEmail author
  • Xiang Xue Li
  • Xue Jia Lai
  • Ke Fei Chen


The paper proposes a type of secret sharing schemes called ramp assignment schemes (RAS’s) to realize general access structures (AS’s). In such a scheme, each participant is assigned a subset of primitive shares of an optimal \((k,L,m)\)-ramp scheme in such a way that the number of primitive shares assigned to each qualified subset is not less than \(k\) whereas the one corresponding to any forbidden subset is not greater than \(k-L\). RAS’s can be viewed as a generalization of multiple assignment schemes (MAS’s). For a same AS, the minimum information rate achieved by MAS’s can not be less than that achieved by RAS’s. With our method, one can find efficient suitable decompositions for general AS’s. And it provides a possibility to refine an existing \((\lambda ,\omega )\)-decomposition. We show that a RAS with minimal worst or/and average information rate can be obtained by linear programming (LP), which is in the complexity class P and much easier than the NP-hard integer programming (IP) applied to constructing optimal MAS’s. Several well-designed algorithms are further presented to cut down the size of the LP/IP problems for optimal RAS’s/MAS’s. Other contributions of the paper include: (1) the current best upper bounds of information rates of two graph AS’s on six participants are improved; (2) some specific AS’s are recognized so that one can obtain the corresponding optimal RAS’s directly, i.e., even without the need of solving LP problems; and (3) we characterize the AS’s of ideal RAS’s and of ideal MAS’s.


Secret sharing (94A62) Ramp assignment scheme Multiple assignment scheme Information rate Linear/integer programming 

Mathematics Subject Classification

94A62 94A60 



The authors appreciate the supports from the National Natural Science Foundation of China under Grant Nos. 61272037, 61070249, 60970111, 61133014 and 60803146.


  1. 1.
    Shamir A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979).Google Scholar
  2. 2.
    Blakley G.R.: Safeguarding cryptographic keys. In: AFIPS 1979. National Computer Conference, New York, vol. 48, pp. 137–313 (1979).Google Scholar
  3. 3.
    Brickell E.F., Stinson D.R.: Some improved bounds on the information rate of perfect secret sharing schemes. J. Cryptol. 5(3), 153–166 (1992).Google Scholar
  4. 4.
    Blundo C.: Secret sharing schemes for access structures based on graphs (in Italian). Tesi di Laurea (1991).Google Scholar
  5. 5.
    Martin K.M.: Discrete structures in the theory of secret sharing. Ph.D. thesis, Royal Holloway and Bedford New College, University of London, London (1991).Google Scholar
  6. 6.
    Martin K.M.: New secret sharing schemes from old. J. Comb. Math. Comb. Comput. 14, 65–77 (1993).Google Scholar
  7. 7.
    Karnin E.D., Greene J.W., Hellman M.E.: On secret sharing systems. IEEE Trans. Inf. Theory 29, 35–41 (1983).Google Scholar
  8. 8.
    Capocelli R.M., Santis A.D., Gargano L., Vaccaro U.: On the size of shares for secret sharing schemes. J. Cryptol. 6(3), 157–167 (1993).Google Scholar
  9. 9.
    Csirmaz L.: The size of a share must be large. J. Cryptol. 10, 223–231 (1997).Google Scholar
  10. 10.
    Brickell E.F.: Some ideal secret sharing schemes. In: EUROCRYPT ’89, Houthalen, vol. 434, pp. 468–475 (1990).Google Scholar
  11. 11.
    Blakley G.R., Meadows C.: Security of ramp schemes. In: CRYPTO’84, Santa Barbara, vol. 196, pp. 242–268 (1985).Google Scholar
  12. 12.
    Yamamoto Y.: On secret sharing systems using \((k, L, n)\)-threshold scheme (in Japanese). Trans. IEICE J68-A(9), 945–952 (1985).Google Scholar
  13. 13.
    Ito M., Saito A., Nishizeki T.: Secret sharing scheme realizing any access structure. In: IEEE Globecom 1987, Tokyo, pp. 99–102 (1987).Google Scholar
  14. 14.
    Benaloh J., Leichter J.: Generalized secret sharing and monotone functions. In: CRYPTO’88, Santa Barbara, No. 403, pp. 25–35 (1988).Google Scholar
  15. 15.
    Stinson D.R.: An explication of secret sharing schemes. Des. Codes Cryptogr. 2, 357–390 (1992).Google Scholar
  16. 16.
    Stinson D.R.: New general lower bounds on the information rate of secret sharing schemes. In: CRYPTO’92, Santa Barbara, No. 740, pp. 168–182 (1993).Google Scholar
  17. 17.
    Stinson D.R.: Decomposition constructions for secret-sharing schemes. IEEE Trans. Inf. Theory 40, 118–125 (1994).Google Scholar
  18. 18.
    Blundo C., Santis A.D., Stinson D.R., Vaccaro U.: Graph decompositions and secret sharing schemes. J. Cryptol. 8, 39–64 (1995).Google Scholar
  19. 19.
    Blundo C., Santis A.D., Gaggia A.G., Vaccaro U.: New bounds on the information rate of secret sharing schemes. IEEE Trans. Inf. Theory 41(2), 549–554 (1995).Google Scholar
  20. 20.
    Dijk M.V.: On the information rate of perfect secret sharing schemes. Des. Codes Cryptogr. 6, 143–169 (1995).Google Scholar
  21. 21.
    Jackson W.A., Martin K.M.: Perfect secret sharing schemes on five participants. Des. Codes Cryptogr. 9, 267–286 (1996).Google Scholar
  22. 22.
    Dijk M.V., Jackson W.A., Martin K.M.: A general decomposition construction for incomplete secret sharing schemes. Des. Codes Cryptogr. 15, 301–321 (1998).Google Scholar
  23. 23.
    Li Q., Yan H., Chen K.F.: A new method of using \((k, n)\)-threshold scheme to realize any access structure (in Chinese). J. Shanghai Jiaotong Univ. 38(1), 103–106 (2004).Google Scholar
  24. 24.
    Iwamoto M., Yamamoto H., Ogawa H.: Optimal multiple assignments based on integer programming in secret sharing schemes. In: ISIT 2004, Chicago, pp. 16–16 (2004).Google Scholar
  25. 25.
    Dijk M.V., Kevenaar T., Schrijen G., Tuyls P.: Improved constructions of secret sharing schemes by applying \((\lambda ,\omega )\)-decompositions. Inf. Process. Lett. 99(4), 154–157 (2006).Google Scholar
  26. 26.
    Iwamoto M., Yamamoto H., Ogawa H.: Optimal multiple assignments based on integer programming in secret sharing schemes with general access structure. Trans. IEICE E90-A(1), 101–112 (2007).Google Scholar
  27. 27.
    Chvátal V.: Linearing Programming. W.H. Freeman, New York (1983).Google Scholar
  28. 28.
    Karmarkar N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984).Google Scholar
  29. 29.
    Garey M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1990).Google Scholar
  30. 30.
    Brickell E.F., Davenport D.M.: On the classification of ideal secret sharing schemes. J. Cryptol. 4(73), 123–134 (1991).Google Scholar
  31. 31.
    Blundo C., De Santis A., Vaccaro U.: Efficient sharing of many secrets. In: STACS’93, Würzburg, vol. 665, pp. 692–703 (1993).Google Scholar
  32. 32.
    Kurosawa K., Okada K., Sakano K., Ogata W., Tsujii T.: Nonperfect secret sharing schemes and matroids. In: EUROCRYPT’93, Lofthus, pp. 126–141 (1993).Google Scholar
  33. 33.
    Simmons G.J., Jackson W.A., Martin K.M.: The geometry of shared secret schemes. Bull. Inst. Comb. Appl. 1, 71–88 (1991).Google Scholar
  34. 34.
    Li Q., Li X.X., Zheng D., Chen K.F.: Optimal multiple assignments with \((m, m)\)-schemes for general access structures. Cryptology ePrint Archive, Report 2012/007,
  35. 35.
    Cormen T.H., Leiserson C.E., Rivest R.L., Stein C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009).Google Scholar
  36. 36.
    Ye Y.: Interior Point Algorithms, Theory and Analysis. Wiley, New York (1997).Google Scholar
  37. 37.
    Jackson W.A., Martin K.M.: A combinatorial interpretation of ramp schemes. Australas. J. Comb. 14, 51–60 (1996).Google Scholar
  38. 38.
    Beimel A., Chor B.: Universally ideal secret sharing schemes. IEEE Trans. Inf. Theory 40(3), 786–794 (1994).Google Scholar
  39. 39.
    Beimel A., Tassa T., Weinreb E.: Characterizing ideal weighted threshold secret sharing. In: TCC’05, Cambridge, vol. 3378, pp. 600–619 (2005).Google Scholar
  40. 40.
    Beimel A., Livne N., Padró C.: Matroids can be far from ideal secret sharing. In: TCC’08, New York, vol. 4948, pp. 194–212 (2008).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Qiang Li
    • 1
    • 4
    Email author
  • Xiang Xue Li
    • 3
    • 4
  • Xue Jia Lai
    • 1
  • Ke Fei Chen
    • 2
  1. 1.School of Electronic Information and Electrical EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.School of ScienceHangzhou Normal UniversityHangzhouChina
  3. 3.Department of Computer Science and TechnologyEast China Normal UniversityShanghaiChina
  4. 4.National Engineering Laboratory for Wireless SecurityXi’anChina

Personalised recommendations