We study internal compression of communication protocols to their internal entropy, which is the entropy of the transcript from the players' perspective. We 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 $\eps > 0$ of a protocol with internal entropy $H^{int}$ and communication complexity $C$ to a protocol with communication at most order $(H^{int}/\eps)^2 \log(\log(C))$.

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. No such example was previously known.

Joint work with Balthazar Bauer (ENS Lyon) and Amir Yehudayoff (Technion).