Composition of Secret Sharing Schemes Sample Clauses

Composition of Secret Sharing Schemes. ‌ ⊆ Let V ′ and V ′′ be F-vector spaces. Let Σ′ = (π0′ , . . . , πn′ ' ) ⊆ (V ′)∗ and Σ′′ = (π0′′, . . . , πn′′'' ) (V ′′)∗ be secret sharing schemes among n′ and n′′ players respectively. − We define a new secret sharing scheme Σ = Σ′[Σ′′] among n := n′ + n′′ 1 players, the composition of Σ′ with Σ′′, which implements the following sharing functionality. To share s ∈ F among n players, first share it using Σ′, so that n′ shares x1′ , . . . , x′n' are obtained. Then use Σ′′ to share x′n' and obtain n′′ shares x′1′, . . . , xn′′'' . The n shares of s in the scheme Σ are x′1, . . . , x′n'—1 and x′1′, . . . , xn′′'' . In other words, in this composition the n′-th player of the scheme Σ′ has been substituted by the set of players of the scheme Σ′′. ∩ { − } ∩ { } ⊆ { } The player sets of Σ′ and Σ′′ can be identified, respectively, with {1, . . . , n′ − 1, p0} and {n′, . . . , n}, where p0 denotes the player of the scheme Σ′ corre- sponding to the linear form πn′ ' . For a set I 1, . . . , n , define I′ := I 1, . . . , n′ 1 and I′′ := I n′, . . . , n . Then I is accepting for Σ if and only if I′ is accepting for Σ′, or I′ p0 is accepting for Σ′ and I′′ is accepting for Σ′′. In particular, if Σ′ has t-privacy then Σ has t-privacy. As to the multiplicativity property, observe that if the scheme Σ′ ' ′ ′ ′′ {1,...,n —1} obtained by removing player p0 from Σ is multiplicative then Σ [Σ ] is always multiplicative. So it makes sense to exclude this case in the proposition below.