Byzantine agreement with homonyms in synchronous systems

We consider here the Byzantine agreement problem in synchronous systems with homonyms. In this model different processes may have the same authenticated identifier. In such a system of n processes sharing a set of l identifiers, we define a distribution of the identifiers as an integer partition of n into l parts n1 ,…, nl giving for each identifier i the number of processes having this identifier. Assuming that the processes know the distribution of identifiers we give a necessary and sufficient condition on the integer partition of n to solve the Byzantine agreement with at most t Byzantine processes. Moreover we prove that there exists a distribution of l identifiers enabling to solve Byzantine agreement with at most t Byzantine processes if and only if $l > \frac{(n-r)t}{n-t-min(t,r)}$ where $r = n \bmod l $ . This bound is to be compared with the l>3t bound proved in [4] when the processes do not know the distribution of identifiers.