# Graph Decompositions and Secret Sharing Schemes

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## Abstract

In this paper, we continue a study of secret sharing schemes for access structures based on graphs. Given a graph *G*, we require that a subset of participants can compute a secret key if they contain an edge of *G*; otherwise, they can obtain no information regarding the key. We study the information rate of such schemes, which measures how much information is being distributed as shares as compared to the size of the secret key, and the average information rate, which is the ratio between the secret size and the arithmetic mean of the size of the shares. We give both upper and lower bounds on the optimal information rate and average information rate that can be obtained. Upper bounds arise by applying entropy arguments due to Capocelli et al [10]. Lower bounds come from constructions that are based on graph decompositions. Application of these constructions requires solving a particular linear programming problem. We prove some general results concerning the information rate and average information rate for paths, cycles and trees. Also, we study the 30 (connected) graphs on at most five vertices, obtaining exact values for the optimal information rate in 26 of the 30 cases, and for the optimal average information rate in 28 of the 30 cases.

## Keywords

Bipartite Graph Connected Graph Linear Programming Problem Secret Sharing Vertex Cover## References

- [1]J. Benaloh and J. Leichter. Generalized secret sharing and monotone functions.
*Lecture Notes in Computer Science*, 403:27–35, 1990.Google Scholar - [2]C. Berge.
*Graphs*. Second revised edition, North-Holland, 1985.Google Scholar - [3]G. R. Blakley. Safeguarding cryptographic keys.
*AFIPS Conference Proceedings*, 48:313–317, 1979.Google Scholar - [4]C. Blundo.
*Secret Sharing Schemes for Access Structures based on Graphs*. Tesi di Laurea, University of Salerno, Italy, 1991, (in Italian).Google Scholar - [5]E. F. Brickell. Some ideal secret sharing schemes.
*J. Combin. Math. and Combin. Comput.*, 9:105–113, 1989.MathSciNetGoogle Scholar - [6]E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes.
*J. Cryptology*, 4:123–134, 1991.zbMATHGoogle Scholar - [7]E. F. Brickell and D. R. Stinson. Some improved bounds on the information rate of perfect secret sharing schemes.
*Lecture Notes in Computer Science*, 537:242–252, 1991. To appear in*J. Cryptology*.Google Scholar - [8]E. F. Brickell and D. R. Stinson. Some improved bounds on the information rate of perfect secret sharing schemes. Department of Computer Science and Engineering Report Series # 106, University of Nebraska, May 1990.Google Scholar
- [9]E. F. Brickell and D. R. Stinson. The detection of cheaters in threshold schemes.
*SIAM J. on Discrete Math.*, 4:502–510, 1991.zbMATHCrossRefMathSciNetGoogle Scholar - [10]R. M. Capocelli, A. De Santis, L. Gargano, and U. Vaccaro. On the size of shares for secret sharing schemes.
*Lecture Notes in Computer Science*, 576:101–113, 1991. To appear in J. Cryptology.Google Scholar - [11]M. R. Garey and D. S. Johnson.
*Computers and Intractability. A Guide to Theory of NP-Completeness*. W. H. Freeman and Company, New York, 1979.zbMATHGoogle Scholar - [12]O. Goldreich, S. Micali, and A. Wigderson. How to play any mental game,
*Proc. 19th ACM Symp. on Theory of Computing*, pages 218–229, 1987.Google Scholar - [13]I. Ingemarsson and G. J. Simmons. A protocol to set up shared secret schemes without the assistance of a mutually trusted party.
*Lecture Notes in Computer Science*, 473:266–282, 1991.Google Scholar - [14]M. Ito, A. Saito, and T. Nishizeki. Secret sharing scheme realizing general access structure.
*Proc. IEEE Globecom’ 87*, pages 99–102, 1987.Google Scholar - [15]E. D. Karnin, J. W. Greene, and M. E. Ilellman. On secret sharing systems.
*IEEE Transactions on Information Theory*, IT-29:35–41, 1983.CrossRefGoogle Scholar - [16]K. M. Martin.
*Discrete Structures in the Theory of Secret Sharing*. PhD Thesis, University of London, 1991.Google Scholar - [17]K. M. Martin. New secret sharing schemes from old. Submitted to
*Journal of Combin. Math. and Combin. Comput.*Google Scholar - [18]R. J. McEliece and D. V. Sarwate. On sharing secrets and Reed-Solomon codes.
*Commun. of the ACM*, 24:583–584, 1981.CrossRefMathSciNetGoogle Scholar - [19]T. Rabin and M. Ben-Or. Verifiable secret sharing and multiparty protocols with honest majority.
*Proc. 21st ACM Symp. on Theory of Computing*, pages 73–85, 1989.Google Scholar - [20]P. D. Seymour. On secret-sharing matroids. Preprint.Google Scholar
- [21]A. Shamir. How to share a secret.
*Commun. of the ACM*, 22:612–613, 1979.zbMATHCrossRefMathSciNetGoogle Scholar - [22]G. J. Simmons. Shared secret and/or shared control schemes.
*Lecture Notes in Computer Science*, 537:216–241, 1991.Google Scholar - [23]G. J. Simmons. Robust shared secret schemes or ‘how to be sure you have the right answer even though you don’t know the question’.
*Congressus Number*., 68:215–248, 1989.MathSciNetGoogle Scholar - [24]G. J. Simmons. How to (really) share a secret.
*Lecture Notes in Computer Science*, 403:390–448, 1990.Google Scholar - [25]G. J. Simmons. An introduction to shared secret and/or shared control schemes and their application.
*Contemporary Cryptology*, IEEE Press, pages 441–497, 1991.Google Scholar - [26]G. J. Simmons. Prepositioned shared secret and/or shared control schemes.
*Lecture Notes in Computer Science*, 434:436–467, 1990.Google Scholar - [27]G. J. Simmons, W. Jackson, and K. Martin. The geometry of shared secret schemes.
*Bulletin of the ICA*, 1:71–88, 1991.zbMATHMathSciNetGoogle Scholar - [28]M. Tompa and H. Woll. How to share a secret with cheaters.
*J. Cryptology*, 1:133–138, 1988.zbMATHCrossRefMathSciNetGoogle Scholar