# Byzantine Generals Problem - Oral and Signed Messages

Having just read through Lamport's paper, I was hoping someone could clarify a few things on the Oral and Signed message algorithms.

1. Why do we have to run $$OM()$$, recursively by $$m$$? If a majority of the Lieutenants are loyal, do they not get the correct value by the first round of exchange?
2. I'm not exactly clear on step 2 of the $$SM()$$ algorithm:
• If Lieutenant $$i$$ receives a message of the form $$v : 0 :j_1 : \ldots : Jk$$
and $$v$$ is not in the set $$V_i$$, then
• he adds v to $$V_i$$;
• if $$k < m$$, then he sends the message $$v:0 :j_1 :\ldots:j_k : i$$ to every
lieutenant other than $$j_1 \ldots j_k$$.

Does Lieutenant $$i$$ only sign a message if $$v$$ is not already in $$V_i$$? Does that mean, there is only one message per value getting $$m$$ signatures on the system? Is that what Lamport means, when they prove that a Lieutenant knows when there are no more messages because there can only be one form of each?

Any tips and help is much appreciated!