Secret Sharing and Non-Shannon Information Inequalities

The known secret-sharing schemes for most access structures are not efficient; even for a one-bit secret the length of the shares in the schemes is 2 O (n ), where n is the number of participants in the access structure. It is a long standing open problem to improve these schemes or prove that they cannot be improved. The best known lower bound is by Csirmaz (J. Cryptology 97), who proved that there exist access structures with n participants such that the size of the share of at least one party is n /logn times the secret size. Csirmaz's proof uses Shannon information inequalities, which were the only information inequalities known when Csirmaz published his result. On the negative side, Csirmaz proved that by only using Shannon information inequalities one cannot prove a lower bound of *** (n ) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. This raises the hope that these inequalities can help in improving the lower bounds beyond n . However, in this paper we show that all the inequalities known to date cannot prove a lower bound of *** (n ) on the share size.

[1]  Brent Waters,et al.  Attribute-based encryption for fine-grained access control of encrypted data , 2006, CCS '06.

[2]  F. Matús,et al.  Two Constructions on Limits of Entropy Functions , 2007, IEEE Transactions on Information Theory.

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

[4]  Brent Waters,et al.  Ciphertext-Policy Attribute-Based Encryption: An Expressive, Efficient, and Provably Secure Realization , 2011, Public Key Cryptography.

[5]  Zhen Zhang,et al.  On Characterization of Entropy Function via Information Inequalities , 1998, IEEE Trans. Inf. Theory.

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

[7]  Avi Wigderson,et al.  Completeness theorems for non-cryptographic fault-tolerant distributed computation , 1988, STOC '88.

[8]  Alex J. Grant,et al.  Dualities Between Entropy Functions and Network Codes , 2008, IEEE Transactions on Information Theory.

[9]  Zhen Zhang.  On a new non-Shannon type information inequality , 2003, Commun. Inf. Syst..

[10]  Alfredo De Santis,et al.  On the Information Rate of Secret Sharing Schemes , 1996, Theor. Comput. Sci..

[11]  Raymond W. Yeung,et al.  A First Course in Information Theory (Information Technology: Transmission, Processing and Storage) , 2006 .

[12]  David Chaum,et al.  Multiparty unconditionally secure protocols , 1988, STOC '88.

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

[14]  Randall Dougherty,et al.  Linear rank inequalities on five or more variables , 2009, ArXiv.

[15]  Amos Beimel,et al.  Universally ideal secret-sharing schemes , 1994, IEEE Trans. Inf. Theory.

[16]  Anna Gál,et al.  Lower bounds for monotone span programs , 1994, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[17]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[18]  Nikolai K. Vereshchagin,et al.  Inequalities for Shannon Entropy and Kolmogorov Complexity , 1997, J. Comput. Syst. Sci..

[19]  Marten van Dijk.  On the information rate of perfect secret sharing schemes , 1995, Des. Codes Cryptogr..

[20]  Alfredo De Santis,et al.  On Secret Sharing Schemes , 1998, Inf. Process. Lett..

[21]  Mitsuru Ito,et al.  Secret sharing scheme realizing general access structure , 1989 .

[22]  Satoru Fujishige,et al.  Polymatroidal Dependence Structure of a Set of Random Variables , 1978, Inf. Control..

[23]  Anna Gál.  A characterization of span program size and improved lower bounds for monotone span programs , 2001, computational complexity.

[24]  Avi Wigderson,et al.  On span programs , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[25]  Randall Dougherty,et al.  Networks, Matroids, and Non-Shannon Information Inequalities , 2007, IEEE Transactions on Information Theory.

[26]  Søren Riis.  Graph Entropy, Network Coding and Guessing games , 2007, ArXiv.

[27]  Alex J. Grant,et al.  The Minimal Set of Ingleton Inequalities , 2011, IEEE Transactions on Information Theory.

[28]  Jessica Ruth Metcalf-Burton.  Improved upper bounds for the information rates of the secret sharing schemes induced by the Vámos matroid , 2011, Discret. Math..

[29]  Raymond W. Yeung,et al.  On a relation between information inequalities and group theory , 2002, IEEE Trans. Inf. Theory.

[30]  László Csirmaz.  The Size of a Share Must Be Large , 1994, EUROCRYPT.

[31]  Frantisek Matús,et al.  Infinitely Many Information Inequalities , 2007, 2007 IEEE International Symposium on Information Theory.

[32]  Nikolai K. Vereshchagin,et al.  A new class of non-Shannon-type inequalities for entropies , 2002, Commun. Inf. Syst..

[33]  Carles Padró,et al.  Matroids Can Be Far from Ideal Secret Sharing , 2008, TCC.

[34]  Weidong Xu,et al.  A projection method for derivation of non-Shannon-type information inequalities , 2008, 2008 IEEE International Symposium on Information Theory.

[35]  Michael O. Rabin,et al.  Randomized byzantine generals , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[36]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[37]  Ueli Maurer,et al.  General Secure Multi-party Computation from any Linear Secret-Sharing Scheme , 2000, EUROCRYPT.

[38]  Mihir Bellare,et al.  Robust computational secret sharing and a unified account of classical secret-sharing goals , 2007, CCS '07.

[39]  Alfredo De Santis,et al.  On the size of shares for secret sharing schemes , 1991, Journal of Cryptology.

[40]  T. H. Chan,et al.  Balanced information inequalities , 2003, IEEE Trans. Inf. Theory.

[41]  Avi Wigderson,et al.  Superpolynomial Lower Bounds for Monotone Span Programs , 1996, Comb..

[42]  Randall Dougherty,et al.  Six New Non-Shannon Information Inequalities , 2006, 2006 IEEE International Symposium on Information Theory.

[43]  Raymond W. Yeung,et al.  A First Course in Information Theory , 2002 .

[44]  Marten van Dijk.  A Linear Construction of Perfect Secret Sharing Schemes , 1994, EUROCRYPT.

[45]  Matthew K. Franklin,et al.  Weakly-Private Secret Sharing Schemes , 2007, TCC.

[46]  G. R. BLAKLEY.  Safeguarding cryptographic keys , 1979, 1979 International Workshop on Managing Requirements Knowledge (MARK).

[47]  Pavel Pudlák.  Monotone complexity and the rank of matrices , 2002, Electron. Colloquium Comput. Complex..

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

[49]  Eyal Kushilevitz,et al.  Secret sharing over infinite domains , 1993, Journal of Cryptology.

[50]  Yvo Desmedt,et al.  Shared Generation of Authenticators and Signatures (Extended Abstract) , 1991, CRYPTO.

[51]  Moni Naor,et al.  Access Control and Signatures via Quorum Secret Sharing , 1998, IEEE Trans. Parallel Distributed Syst..