Framing in Secret Sharing

Secret sharing, a well-known cryptographic technique, introduced 40 years ago as a private and reliable variant of classical storage, has now become a major cryptographic primitive with numerous real-world applications. In this paper we consider the digital forensics aspects of secret sharing. We investigate the problem of framing which occurs when a coalition is able to calculate the share of a participant who does not belong to it. In the extreme case one authorized coalition can calculate shares of another authorized coalition and use the secret in some way blaming another authorized coalition for their action. In this context seniority plays an important role. We define seniority, which comes natural in the context of hierarchical access structures. Roughly speaking, our work shows that in an ideal secret sharing scheme an authorized coalition cannot frame participants who are less senior than all members of the coalition and is able to frame a participant who is more senior than at least one pivotal member of the coalition. We show that for any monotone access structure there exists a (non-ideal) frameproof secret sharing scheme.

[1]  Yvo Desmedt,et al.  Threshold Cryptosystems , 1989, CRYPTO.

[2]  Yvo Desmedt,et al.  Threshold cryptography , 1994, Eur. Trans. Telecommun..

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

[4]  Caroline J. Klivans.  Threshold graphs, shifted complexes, and graphical complexes , 2007, Discret. Math..

[5]  Alfred Lehman,et al.  A Solution of the Shannon Switching Game , 1964 .

[6]  James G. Oxley,et al.  Matroid theory , 1992 .

[7]  Tatiana Gvozdeva,et al.  Weightedness and structural characterization of hierarchical simple games , 2013, Math. Soc. Sci..

[8]  Kouya Tochikubo,et al.  New Secret Sharing Schemes Realizing General Access Structures , 2015, J. Inf. Process..

[9]  Paul D. Seymour.  On secret-sharing matroids , 1992, J. Comb. Theory, Ser. B.

[10]  Josep Freixas,et al.  Complete simple games , 1996 .

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

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

[13]  Tamir Tassa,et al.  Hierarchical Threshold Secret Sharing , 2004, Journal of Cryptology.

[14]  Andrew Chi-Chih Yao,et al.  Protocols for secure computations , 1982, FOCS 1982.

[15]  Carles Padró,et al.  Ideal Hierarchical Secret Sharing Schemes , 2010, IEEE Transactions on Information Theory.

[16]  Yvo Desmedt,et al.  Realistic versus Rational Secret Sharing , 2019, GameSec.

[17]  Dan Boneh,et al.  Collusion-Secure Fingerprinting for Digital Data , 1998, IEEE Trans. Inf. Theory.

[18]  Moti Yung,et al.  Perfectly secure message transmission , 1993, JACM.

[19]  Peter Sudhölter,et al.  Directed and weighted majority games , 1995, Math. Methods Oper. Res..

[20]  Ernest F. Brickell,et al.  Some Ideal Secret Sharing Schemes , 1990, EUROCRYPT.

[21]  Piotr Faliszewski,et al.  Cloning in Elections: Finding the Possible Winners , 2011, J. Artif. Intell. Res..

[22]  Amos Fiat,et al.  Tracing traitors , 2000, IEEE Trans. Inf. Theory.

[23]  Josh Benaloh,et al.  Generalized Secret Sharing and Monotone Functions , 1990, CRYPTO.

[24]  Douglas R. Stinson,et al.  Anonymous Secret Sharing Schemes , 1997, Discret. Appl. Math..

[25]  Gustavus J. Simmons,et al.  How to (Really) Share a Secret , 1988, CRYPTO.

[26]  Alan D. Taylor,et al.  Simple Games , 2000 .

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

[28]  Yongge Wang,et al.  Perfectly Secure Message Transmission Revisited , 2002, IEEE Transactions on Information Theory.

[29]  C. L. Liu,et al.  Introduction to Combinatorial Mathematics. , 1971 .

[30]  Ernest F. Brickell,et al.  On the classification of ideal secret sharing schemes , 1989, Journal of Cryptology.

[31]  Gustavus J. Simmons,et al.  Prepositioned Shared Secret and/or Shared Control Schemes , 1990, EUROCRYPT.