We investigate the combinatorial properties of threshold schemes. Informally, a (t, w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key, but no subset of fewer than t participants can determine the key. Our interest is in perfect threshold schemes: no subset of fewer than t participants can determine any partial information regarding the key. We give a combinatorial characterization of a certain type of perfect threshold scheme. We also investigate the maximum number of keys which a perfect (t, w)-threshold scheme can incorporate, as a function of t, w, and the total number of possible shadows, v. This maximum can be attained when there is a Steiner system S(t, w, v) which can be partitioned into Steiner systems S(t − 1. w, v). Using known constructions for such Steiner systems, we present two new classes of perfect threshold schemes, and discuss their implementation.
[1]
Suresh C. Kothari,et al.
Generalized Linear Threshold Scheme
,
1985,
CRYPTO.
[2]
Catherine A. Meadows,et al.
Security of Ramp Schemes
,
1985,
CRYPTO.
[3]
Albrecht Beutelspacher,et al.
Geometric Structures as Threshold Schemes
,
1986,
EUROCRYPT.
[4]
Richard M. Wilson.
Some partitions of all triples into steiner triple systems
,
1974
.
[5]
G. R. BLAKLEY.
Safeguarding cryptographic keys
,
1979,
1979 International Workshop on Managing Requirements Knowledge (MARK).
[6]
Ian Holyer,et al.
The NP-Completeness of Edge-Coloring
,
1981,
SIAM J. Comput..
[7]
Adi Shamir,et al.
How to share a secret
,
1979,
CACM.
[8]
Ronald D. Baker,et al.
Partitioning the planes of AG2m(2) into 2-designs
,
1976,
Discret. Math..