Question on proof of coin-tossing in "how to simulate it"

Question

I am reading Professor Lindell's "How To Simulate It - A Tutorial on the Simulation Proof Technique" paper, which is very enlightening.

I am trying to follow through the proof of the security of Blum's coin-tossing protocol and have some questions: in the end of page 41 it says ": Let $b_1$ be the value committed in the commitment $c$ sent by $A$ (since $A$ is deterministic and this is the first message, this is a fixed value)."

1. I don't understand why we can assume $A$ is deterministic?
2. In general, should the security proof work for probabilistic adversaries?

Answer 1

Actually, a probabilistic adversary $A$ can be viewed as a deterministic adversary with a random tape: $A$ just picks a random string in the beginning and then run deterministically according to that fixed string. In the proof, the simulator $S$ uses the real-world adversary $A$ as a black box but can control its random tape (because a random tape always contains fair random strings that can be generated by anyone). So, $S$ can simulate the output of a probabilistic adversary $A$ by first choosing a random string $r$ then dealing with the resulting deterministic adversary $A'=A(r)$.