Multi-Party Protocols, Information Complexity and Privacy

TL;DR

This paper introduces Public Information Complexity (PIC), a new measure for analyzing multi-party protocols, providing bounds on communication, randomness, and enabling protocol compression, with explicit calculations for the parity function.

Contribution

The paper defines PIC and demonstrates its properties, including bounds on communication and randomness, and applies it to compute the complexity of the parity function in multi-party settings.

Findings

PIC is a lower bound on communication complexity.

PIC is an upper bound on information complexity.

The randomness needed for parity computation is at least proportional to input size.

Abstract

We introduce a new information theoretic measure that 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; an explicit calculation of the zero-error Public Information Complexity of the $k$-party, $n$-bit Parity function, where a player outputs the bit-wise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is $\Omega(n)$.