Security Proof F Sample Clauses
Security Proof
F. After describing our A-OCC protocol, we proceed to present and prove the formal security statement that demonstrates how our A-OCC protocol UC-realizes a-occ. Next, we prove a combinatorial observation regarding vectors of random values. Finally, we utilize this combinatorial observation 13Feldman [Fel89] calculated the size of the overlap, denoted as x, based on the number of participants n and the maximum number of corruptions t. The general relation is x ≥ n − t − t2 , which yields x ≥ n/3 and x ≥ 5n/8 when t ≤ n/3 and t ≤ n/4, respectively. n−2t Protocol ΠVa-occ The protocol is parameterized by a set V of possible outcomes. Let m = lcm(n2, |V |). Upon receiving input (input, sid) from the environment, where sid = (P, sid′) for a player set P of size n, and its following activations, party Pi ∈ P proceeds as follows.
1. Initialize sets Ci, Gi, Zi, and Ri to 𝜙, flag finished to 0, and variable yi to ⊥. Also initialize Cj′ .= ⊥, G′j .= ⊥, bj,k .= 0, rk,j .= ⊥, vj .= ⊥, and zj′ .= ⊥ for all j, k ∈ [n].
2. For each j ∈ [n], choose x R [m] and send (share, sidshare,j, x ) to an instance of F F (F is the ▇▇▇▇▇▇,▇ . a-vss smallest prime field with size at least m) with SID sidi = (P, Pi, sid , share, j). Upon later activations from the environment, if finished = 0 then do:
1. For each j, k ∈ [n] such that bj,k = 0, in a round-▇▇▇▇▇ fashion (across activations) fetch the share from the instance of FaF-vss with ▇▇▇ ▇▇▇▇▇▇▇▇,k. Upon receiving back (shared, sidshare,k) set bj,k .= 1, and if bj,l = 1 for all l ∈ [n] then update Ci .= Ci ∪ {Pj}.
2. Wait until |Ci| ≥ t+1. Then, send (send, sidattach, C(t+1)) to an instance of Fa-cast with ▇▇▇ ▇▇▇▇▇▇▇▇▇ .=
i i (P, Pi, sid′, attach).
3. For each j ∈ [n] such that Cj′ = ⊥, in a round-▇▇▇▇▇ fashion fetch the output from the instance of Fa-cast with ▇▇▇ ▇▇▇▇▇▇▇▇▇. Upon receiving back (output, sidattach, Y ), record C′ .= Y , and if C′ ⊆ Ci then update .=