Size of Shares and Probability of Cheating in Threshold Schemes

In this paper we study the amount of secret information that must be given to participants in any secret sharing scheme that is secure against coalitions of dishonest participants in the model of Tompa and Woll [20]. We show that any (k, n) threshold secret sharing algorithm in which any coalition of less than k participants has probability of successful cheating less than some ? > 0 it must give to each participant shares whose sizes are at least the size of the secret plus log 1/?.

[1]  Douglas R. Stinson,et al.  Decomposition constructions for secret-sharing schemes , 1994, IEEE Trans. Inf. Theory.

[2]  Douglas R. Stinson,et al.  An explication of secret sharing schemes , 1992, Des. Codes Cryptogr..

[3]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[4]  Douglas R Stinson,et al.  Some improved bounds on the information rate of perfect secret sharing schemes , 1990, Journal of Cryptology.

[5]  I. Csiszár.  Information Theory , 1981 .

[6]  Dorothy E. Denning,et al.  Cryptography and Data Security , 1982 .

[7]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[8]  Alfredo De Santis,et al.  On the Information Rate of Secret Sharing Schemes (Extended Abstract) , 1992, CRYPTO.

[9]  Tal Rabin,et al.  Verifiable secret sharing and multiparty protocols with honest majority , 1989, STOC '89.

[10]  Ernest F. Brickell,et al.  The Detection of Cheaters in Threshold Schemes , 1990, SIAM J. Discret. Math..

[11]  R. F.,et al.  Mathematical Statistics , 1944, Nature.

[12]  R. J. McEliece,et al.  On sharing secrets and Reed-Solomon codes , 1981, CACM.

[13]  G. R. Blakley,et al.  Safeguarding cryptographic keys , 1899, 1979 International Workshop on Managing Requirements Knowledge (MARK).

[14]  Ehud D. Karnin,et al.  On secret sharing systems , 1983, IEEE Trans. Inf. Theory.

[15]  Gustavus J. Simmons,et al.  An Introduction to Shared Secret and/or Shared Control Schemes and Their ApplicationThis work was performed at Sandia National Laboratories and supported by the U.S. Department of Energy under contract number DEAC0476DPOO789. , 1992 .