Efficient Resilient Functions

An n-bit boolean function is resilient to coalitions of size q if no fixed set of q bits is likely to influence the value of the function when the other n - q bits are chosen uniformly at random, even though the function is nearly balanced. We construct explicit functions resilient to coalitions of size q = n/(log n)(O(log log n)) = n(1-o)(1) computable by linear-size circuits and linear-time algorithms. We also obtain a tight size-depth tradeoff for computing such resilient functions. Constructions such as ours were not available even non-explicitly. It was known that functions resilient to coalitions of size q = n(0.63)... can be computed by linear-size circuits [BL85], and functions resilient to coalitions of size q = Theta(n/log(2) n) can be computed by quadratic-size circuits [AL93]. One component of our proofs is a new composition theorem for resilient functions.