Multipartite Secret Sharing by Bivariate Interpolation

Given a set of participants that is partitioned into distinct compartments, a multipartite access structure is an access structure that does not distinguish between participants belonging to the same compartment. We examine here three types of such access structures: two that were studied before, compartmented access structures and hierarchical threshold access structures, and a new type of compartmented access structures that we present herein. We design ideal perfect secret sharing schemes for these types of access structures that are based on bivariate interpolation. The secret sharing schemes for the two types of compartmented access structures are based on bivariate Lagrange interpolation with data on parallel lines. The secret sharing scheme for the hierarchical threshold access structures is based on bivariate Lagrange interpolation with data on lines in general position. The main novelty of this paper is the introduction of bivariate Lagrange interpolation and its potential power in designing schemes for multipartite settings, as different compartments may be associated with different lines or curves in the plane. In particular, we show that the introduction of a second dimension may create the same hierarchical effect as polynomial derivatives and Birkhoff interpolation were shown to do in Tassa (J. Cryptol. 20:237–264, 2007).

[1]  K. Chung,et al.  On Lattices Admitting Unique Lagrange Interpolations , 1977 .

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

[3]  Anna Gál,et al.  Combinatorial methods in boolean function complexity , 1995 .

[4]  Moni Naor,et al.  Secure and Efficient Metering , 1998, EUROCRYPT.

[5]  Michael J. Collins.  A Note on Ideal Tripartite Access Structures , 2002, IACR Cryptol. ePrint Arch..

[6]  Nira Dyn,et al.  Polynomial Interpolation to Data on Flats in Rd , 2000 .

[7]  Tamir Tassa,et al.  Characterizing Ideal Weighted Threshold Secret Sharing , 2008, SIAM J. Discret. Math..

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

[9]  Serge Fehr.  Ecien t Construction of the Dual Span Program , 1999 .

[10]  Carles Padró,et al.  Secret Sharing Schemes with Bipartite Access Structure , 1998, EUROCRYPT.

[11]  Germán Sáez,et al.  New Results on Multipartite Access Structures , 2006, IACR Cryptol. ePrint Arch..

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

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

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

[15]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[16]  Tamir Tassa.  Hierarchical Threshold Secret Sharing , 2004, TCC.

[17]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.