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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.


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It is safe, in the case that you have MACs which are independently keyed (or at the very least, the cryptographically secure MAC is independently keyed from all the other ones). This can be seen not by an argument from randomness, but from a simple observation that it were not true, then an attacker could attack the secure MAC by generating the insecure ...



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