Each value Vi1, Sample Clauses

Each value Vi1,. Vi(n−2t) is distributed uniformly over [0, . . . , u] and independent of the values associated with the other parties. that Pk ∈ M implies that Pk belongs to 55 of at least t + 1 parties P5, who are present in £i. This ensures that there is at least one such P5 who belongs to £j, as well as £i. Now P5 ∈ £j implies that Pj had ensured that 55 ⊆ 5j. This im- plies that Pk ∈ M belongs to 5j before the party Pj broad- PROOF: The values Vi1 , . . . , V i(n−2t) are defined in the step casted the Reconstruct Enabled message. Since Hj ∈ is the instance of 5j at the time when Pj broadcasted the ⊆ f ∈ 5 ∈ ⊆ f ∈ 5 10 of the protocol. We now prove the first part of the lemma. According to the lemma condition, Pi j. This implies that Ti j. So the honest Pj will participate in the in- stance Xxxx, corresponding to each Pk Ti. Moreover, eventually Ti m and Xx m will hold for every other honest party Pm. So, every other honest party will also even- tually participate in the instance Xxxx corresponding to each Pk ∈ Ti. Now by the property of Rec, each honest party will Reconstruct Enabled message, it is obvious that Pk M belongs to Hj also. Using similar argument, it can be shown that every Pk M also belongs to Hj, thus proving the second part of the lemma. ∈ | | ≥ To complete the lemma, it remains to show that M |£ | n , for which we use a counting argument. Let m = i at the time when Pi broadcasted the Reconstruct Enabled message. So we have m ≥ 2t+1. Now consider an n×n ta- eventually reconstruct S→k = (xk1, . . . , xxx, . . . , xkn) at the completion of Rec , except with probability ϵ′. Thus, with ble Xx (relative to party Pi), whose lth row and kth column — — ≈ — probability 1 (n t)ϵ′ 1 ϵ, every honest party will correctly have the vector Xi during the step 10 and hence will associate the values Vi1, . . . , Vi(n−2t) with Pi. — — We now prove second part of the lemma. By Xxxxx 13, when Ti is fixed, the values that are shared by corrupted par- ties in Ti are completely independent of the values shared by the honest parties in Ti. Now Ti contains n t parties and hence at least n 2t honest parties and every honest partys’ shared secrets are uniformly distributed and mutually inde- pendent. This implies that in the vector Xi, there are at least contains 1 for k, l 1, . . . , n if and only if the following holds: (a) Pi has received the message (P5 Accepts G5) from the broadcast of P5 and included P5 in i before broad- casting the Reconstruct Enabled message and (b)Pk G5. ...