Multidimensional Graded Consensus Protocol Sample Clauses

Multidimensional Graded Consensus Protocol. ‌ The notion of Graded Consensus protocol, introduced by Micali in [3], is much weaker than Byzantine agreement but allows the protocol players to gain some information about the distribution of the input values possessed by the network participants. We now provide the definition of an extension of the concept of (n, t)-Graded Consensus to the multidimensional case. 2.1 (m-Dimensional (n, t)-Graded Consensus Protocol ). Let P be a protocol in which the set of all players is common knowledge, and each player i privately knows an arbitrary initial vector of messages v′ = (v′ , v′ , . . . , v′ ) where v′ ∈ V ∪ {⊥}. We say that P is an m-dimensional (n, t)-graded consensus protocol if, in every execution with n players of which at most t are malicious, every honest player i halts outputting a vector of value-grade pairs oi = (vi, gi) = ((xx,0, xx,0), (xx,0, xx,0), . . . , (vi,m, gi,m)) where gi,c ∈ {0, 1, 2} and vi,c ∈ V ∪ {⊥} for every i and c, so as to satisfy the following three conditions: 1. For all honest players i and j, for all c ∈ {1, . . . , m}, we have that |gi,c − gj,c| ≤ 1. 2. For all honest players i and j, for all c ∈ {1, . . . , m} and for all positive gi,c, gj,c we have vi,c = vj,c ⊥. 0. If v′ ′ = v 2,c = · · · = v′ = vc for some value vc ∈ V ∪ {⊥}, then for all honest players the output component c is (vi,c, gi,c) = (vc, 2) if vc ⊥, (vi,c, gi,c) = (⊥, 0) if vc = ⊥ .