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The probability that a probabilistic polynomial adversary corrupting the sender can finds two pairs $(m,r)$ such that the output of the random oracle $c$ is the same (break the binding property) is negligible. At the same time, the probability that a probabilistic polynomial adversary corrupting the sender finds a pair $(m,r)$ starting from the commitment $c$ (break the hiding property) is negligible. Is there any difference between the case in which the adversary is semi-honest or active in this context? The only threat I see is that an active adversary could change the message of the sender, but this is not relevant since the sender could have chosen that message.

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There is no active attack an adversary could do, so the random oracle commitment is secure in (both the passive and) the active adversary model.

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