Optimality of ΠHBA and ΠETHBA Sample Clauses

Optimality of ΠHBA and ΠETHBA. ‌ In this section, we show that ΠHBA and ΠETHBA (presented in the following section) achieve optimal parameters. Concretely, we show that it is not possible to obtain a hybrid protocol which achieves (p, fAR)-output responsiveness and fAV-validity if fAV > 1 (1 − fAR). We also show that it is possible to convert any protocol ΠABA which achieves binary byzantine agreement BABA (with p-termination) when less than fAR ≤ n parties are corrupted into a protocol Πopt that achieves (p, fAR)-termination, fAR-consistency, and 1 (1 − fAR)-validity. This transformation can be used to transform an existing BABA protocol into a protocol that gives optimal parameters when plugged into ΠHBA or ΠETHBA. Depending on whether (output)-responsiveness or validity are prioritized, it is possible to set the parameters in Πopt accordingly (by setting fAR in Πopt to the desired value). The transformation Πopt BABA BABA BABA is described in Figure 4.2. Figure 4.2: Πopt BABA protocol (view of Pi), with parameter fAR. 1. Let bi denote the input of party Pi. 2. Pi computes a signature σi ← Sign(bi, ski) and broadcasts (i, bi, σi) to every party (including itself). 3. Pi wait until it obtains n(1 − fAR) valid messages (i.e., with a valid signature of bi under pki) of the form (i, bi, σi) (from n(1 − fAR) different parties). 4. Let b denote the majority bit among the valid messages that Pi received. Pi broadcasts a message of the form (i, b, Li) to every party (including itself), where the list Li contains all the valid signatures that Pi received on b. 5. Pi runs ΠABA with input b. Let b∗ denote the output of ΠABA. 6. Upon receiving a valid message of the form (j, b∗, Lj) (i.e., where Li contains at least n(1 − fAR) valid signatures on b∗ from different parties), Pi terminates the protocol with output b∗.