On the Power of 1-way Functions

The earliest definition of 1-way function is due to Berman [Ber77], who considered polynomial-time computable, length-increasing, 1–1 functions that do not have a polynomial-time computable inverses. Recently, more powerful notions are considered, e.g., polynomial-time computable, length-increasing, 1–1 functions f such that the probability that a BPP algorithm can compute z from f(x) for a randomly selected x is superpolynomidly small [CYa82]. Whatever definition is used, these functions are necessarily easy invert on some inputs:Proposition 1If f is a polynomial-time computable, length-increasing, 1-1 function, and if p is a polynomial, then there is a polynomial time algorithm that for sufficiently large n inverts f on at least p(n) strings of length less than n. Therefore, the range of every such function must contain a polynomial-time computable subset of arbitrarily large polynomial census.