One-way functions and circuit complexity

Abstract

A finite function f is a mapping of {0,1}n into {0,1}m∪{#}, where “#” is a symbol to be thought of as “undefined.” A family of finite functions is said to be one-way (in a circuit complexity sense) if it can be computed with polynomial size circuits, but no family of inverses of these functions can be computed with polynomial size circuits. In this paper we show that (provided functions that are not one-to-one are allowed) one-way functions exist if and only if the satisfiability problem SAT does not have polynomial size circuits. A family of functions fi(x) can be checked if some family of polynomial size circuits with inputs x and y can determine if fi(x)=y. A family of functions fi(x) can be evaluated if some family of polynomial size circuits with input x can compute fi(x). Can all families of total functions that can be checked also be evaluated? We show that this is true if and only if the nonuniform versions of the complexity classes P and UP∩co-UP are equal. A family of functions fi is one-way for constant depth circuits if fi can be computed with unbounded fan-in circuits of polynomial size and constant depth, but every family of inverses fi−1 cannot. In this paper we give two provably one-way functions (in fact permutations) for constant depth circuits. The second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth.