Bounds on tradeoffs between randomness and communication complexity

A quantitative investigation of the power of randomness in the context of communication complexity is initiated. The authors prove general lower bounds on the length of the random input of parties computing a function f, depending on the number of bits communicated and the deterministic communication complexity of f. Four standard models for communication complexity are considered: the random input of the parties may be shared or local, and the communication may be one-way or two-way. The bounds are shown to be tight for all the models, for all values of the deterministic communication complexity, and for all possible quantities of bits exchanged. It is shown that it is possible to reduce the number of random bits required by any protocol, without increasing the number of bits exchanged (up to a limit depending on the advantage achieved by the protocol)