We precisely characterize the role of private randomness in the ability of Alice to send a message to Bob while minimizing the amount of information revealed to him. We give an example of a (randomized) message which can be transmitted while revealing only I bits of information using private randomness, but requires Alice to reveal I+log I-O(1) bits of information if only public coins are allowed. This gives the first example of an w(l) additive separation between these two models. Our example also shows that the one-round compression construction of Har-sha et al. [HJMR07] cannot be improved. Moreover, we show that our example is tight up to an additive O(1) factor: We show that if using private randomness a message can be transmitted while revealing I bits of information, the transmission can be simulated without private coins using J + log/ + O(1) bits of information. This improves over an earlier result by Brody et al. [BBK~+12].
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