ON THE CIRCUIT-SIZE OF INVERSES

Abstract

We reprove a result of Boppana and Lagarias: If [Formula: see text] then there exists a partial function f that is computable by a polynomial-size family of circuits, but no inverse of f is computable by a polynomial-size family of circuits. We strengthen this result by showing, if [Formula: see text], that there exist length-preserving total functions that are one-way by circuit size and that are computable in uniform polynomial time. We also prove, if [Formula: see text], that there exist polynomially balanced total surjective functions that are one-way by circuit size; here non-uniformity is used.