Multi-Party Protocols, Information Complexity and Privacy.

We introduce a new information-theoretic measure, which we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion: The Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; and an explicit calculation of the zero-error Public Information Complexity of the k-party, n-bit Parity function, where a player outputs the bitwise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is Ω (n).