One- Way Functions and Circuit Complexity

A finite function f is a mapping of {1 , 2 ,. .. , m } into {1 , 2 ,. .. , m } ∪ { # } where # is a symbol to be thought of as ''undefined.'' This paper defines a measure M(f) of the difficulty of inverting a finite function f, which is given by M(f) = MIN    log 2 C(f) log 2 C(g) _ _________ : g an inverse of f    where C(f) is a circuit complexity measure of the difficulty of computing f. We say that one-way functions exist (in a circuit complexity sense) if and only if M(f) is unbounded. We prove that one-way functions exist if and only if the satisfiability problem SAT has polynomial sized circuits. This paper also defines an analogous measure M d (f) in which only circuits of depth ≤ d are allowed. We show that one-way functions exist in this bounded-depth circuit complexity model, by showing for the permutations σ n on {1 , 2 ,. .. , 2 n } defined by σ n (k) ≡ 3k (mod 2 n) that for d ≥ 4 there is a positive constant c d such that M d (σ n) > c d log n as n → ∞.