Private vs. Common Random Bits in Communication Complexity

We investigate the relative power of the common random string model vs. the private random string model in communication complexity. We show that the two model are essentially equal. Communication complexity is a model of computation where two parties, each with an input, want to mutually compute a Boolean function that is deened on pairs of inputs. Formally, let f : X Y 7 ! f0; 1g be a Boolean function. The communication problem for f is the following two-player game. Player A gets x 2 X and player B gets y 2 Y. Their goal is to compute f(x; y). They have unlimited computational power and a full description of f, but they don't know each other's input. They determine the output value by exchanging messages. Let n, the length of the input, be log(jXjjY j). A protocol for computing f is a pair of algorithms (one for each player) according to which the players send binary messages. A protocol proceeds in rounds. In every round the protocol speciies which player's turn it is to send a message. Each player in his turn sends a bit message that may depend on