A Theorem on Sensitivity and Applications in Private Computation

In this paper we prove a theorem that gives an (almost) tight upper bound on the sensitivity of a multiple-output Boolean function in terms of the sensitivity of its coordinates and the size of the range of the function. We apply this theorem to get improved lower bounds on the time (number of rounds) to compute Boolean functions by private protocols. These bounds are given in terms of the sensitivity of the function being computed and the amount of randomness used by the private protocol. These lower bounds are tight (up to constant factors) for the case of the xor function and together with the results in [E. Kushilevitz and A. Rosén, SIAM J. Discrete Math., 11 (1998), pp. 61--80.] establish a tight (up to constant factors) tradeoff between randomness and time in private computation.