Homonyms with forgeable identifiers

We consider here the Byzantine Agreement problem (BA) in synchronous systems with homonyms in the case where some identifiers may be forgeable. More precisely, the n processes share a set of l (1≤l≤n) identifiers. Assuming that at most t processes may be Byzantine and at most k (t≤k≤l) of these identifiers are forgeable in the sense that any Byzantine process can falsely use them, we prove that Byzantine Agreement problem is solvable if and only if l>2t+k. Moreover we extend this result to systems with authentication by signatures in which at most k signatures are forgeable and we prove that Byzantine Agreement problem is solvable if and only if l>t+k.