2) Condition changes for both leader/committee election. This should have been explicit.

3) Any reasonable interpretation here is accepted. Most naturally it means the attacker gets to decide exactly which blocks go into the blockchain. You can show with probability one (in expected constant number of rounds) or with very high probability either is accepted. ]]>

1) I'm a bit confused with the terminology. When an attacker joins **before** round r, is it possible that he already knows N(r) (with probability of atleast 99%) by the time he joins? For example, can we assume that if he publishes his pk just before the beginning of round r, with probability of atleast 99%, no more than m new nodes join after him before round r (where m is constant)?

2) In section b, is the leader chosen according to the original Algorand's consensus, or does the condition change for the leader as well?

3) Also in section b. What does "takes control of the entire system" mean? Does it mean that the attacker is able to control who becomes leader and which nodes will be on the committee for all future rounds? Does it have to be with probability 1?