Lattice-Based Threshold Changeability for Standard Shamir Secret-Sharing Schemes

We consider the problem of increasing the threshold parameter of a secret-sharing scheme after the setup (share distribution) phase, without further communication between the dealer and the shareholders. Previous solutions to this problem require one to start off with a nonstandard scheme designed specifically for this purpose, or to have communication between shareholders. In contrast, we show how to increase the threshold parameter of the standard Shamir secret-sharing scheme without communication between the shareholders. Our technique can thus be applied to existing Shamir schemes even if they were set up without consideration to future threshold increases. Our method is a new positive cryptographic application for lattice reduction algorithms, inspired by recent work on lattice-based list decoding of Reed-Solomon codes with noise bounded in the Lee norm. We use fundamental results from the theory of lattices (geometry of numbers) to prove quantitative statements about the information-theoretic security of our construction. These lattice-based security proof techniques may be of independent interest.

[1]  Ravi Kumar,et al.  A sieve algorithm for the shortest lattice vector problem , 2001, STOC '01.

[2]  Atsuko Miyaji,et al.  Efficient and Unconditionally Secure Verifiable Threshold Changeable Scheme , 2001, ACISP.

[3]  Hugo Krawczyk,et al.  Proactive Secret Sharing Or: How to Cope With Perpetual Leakage , 1995, CRYPTO.

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

[5]  Ueli Maurer,et al.  Generalized privacy amplification , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[6]  Igor E. Shparlinski,et al.  Sparse polynomial approximation in finite fields , 2001, STOC '01.

[7]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[8]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[9]  Igor E. Shparlinski,et al.  Noisy Chinese remaindering in the Lee norm , 2004, J. Complex..

[10]  John Bloom,et al.  A modular approach to key safeguarding , 1983, IEEE Trans. Inf. Theory.

[11]  Sushil Jajodia,et al.  Redistributing Secret Shares to New Access Structures and Its Applications , 1997 .

[12]  Keith M. Martin,et al.  Updating the parameters of a threshold scheme by minimal broadcast , 2005, IEEE Transactions on Information Theory.

[13]  Dana Ron,et al.  Chinese remaindering with errors , 1999, STOC '99.

[14]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[15]  Reihaneh Safavi-Naini,et al.  Bounds and Techniques for Efficient Redistribution of Secret Shares to New Access Structures , 1999, Comput. J..

[16]  Alfredo De Santis,et al.  Fully Dynamic Secret Sharing Schemes , 1993, Theor. Comput. Sci..

[17]  P. Shiu,et al.  Geometric and analytic number theory , 1991 .

[18]  Luisa Gargano,et al.  A Note on Secret Sharing Schemes , 1993 .

[19]  Dan Boneh,et al.  Finding smooth integers in short intervals using CRT decoding , 2000, STOC '00.

[20]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[21]  Bart Preneel,et al.  On the Security of the Threshold Scheme Based on the Chinese Remainder Theorem , 2002, Public Key Cryptography.

[22]  Amin Shokrollahi,et al.  List Decoding of Algebraic-Geometric Codes , 1999, IEEE Trans. Inf. Theory.

[23]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[24]  Josef Pieprzyk,et al.  Changing Thresholds in the Absence of Secure Channels , 1999, Aust. Comput. J..

[25]  Maurice Mignotte,et al.  How to Share a Secret? , 1982, EUROCRYPT.

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

[27]  R. Kannan.  ALGORITHMIC GEOMETRY OF NUMBERS , 1987 .

[28]  Keith M. Martin,et al.  A combinatorial interpretation of ramp schemes , 1996, Australas. J Comb..