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Game-based security notion and hybrid proofs - adversarial randomness

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Say we have a game-based definition of security and wish to prove the security of our construction using a series of hybrids. Can we somehow "fix" an adversary that we interact with in one of the hybrids and continue using it in the following hybrid steps? The answer to this Security proof. Adversary source of random coins. question states that in the game-based proofs it is indeed possible to fix the adversarial random tape to a uniformly chosen string and then work with an effectively deterministic adversary. Why is this meaningful?

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