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Internal compression of protocols to entropy

We study internal compression of communication protocols to their internal entropy, which is the entropy of the transcript from the players’ perspective. We first show that errorless compression to the internal entropy (and hence to the internal information) is impossible. We then provide two internal compression schemes with error. One of a protocol of Fiege et al. for finding the first difference between two strings. The second and main one is an internal compression with error e > 0 of a protocol with internal entropy H int and communication complexity C to a protocol with communication at most order (H int/e)2 log(log(C)). This immediately implies a similar compression to the internal information of public coin protocols, which exponentially improves over previously known public coin compressions in the dependence on C. It further shows that in a recent protocol of Ganor, Kol and Raz it is impossible to move the private randomness to be public without an exponential cost. To the best of our knowledge, no such example was previously known.