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Memory of the first round can't help the attacker win the second round. If it could: the attacker could simulate the first round on his own (picking his own key, pretending to play the game against himself), and then play the second round against the challenger. So, adversaries like this are not stronger -- not even if they have memory.


You always need to have in mind that $A$ is a hypothetical algorithm, since our goal in the reduction is to contradict the existence of such an efficient $A$. Now to your concrete security framework: Here, you are not satisfied by the fact that a hypothetical poly-time $A$ implies a poly-time reduction $A'$, but your aim is that the reduction does not take ...


The equation $t'=t+n\cdot t_c$ is an estimation to put an upper limit on $t'$. It might be possible that an attacker $A'$ can use a different, more efficient algorithm. But since the attack will work with using $A$, there exists an attacker $A'$ with at most $t'$. This means it's actually not an equation, but an inequality $t' \leq t + n \cdot t_c$. And then ...

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