Constructions of feebly-one-way families of permutations

Abstract

The unrestricted circuit complexity C(·) over the basis of all logic 2-input/1-output gates is considered. It is proved that certain explicitly defined families of permutations {fn} are feebly-one-way of order 2, i.e., the functions fn satisfy the property that, for increasing n, C(fn-1) approaches 2 · C(fn) while C(fn) tends to infinity. Both these functions and their corresponding complexities are derived by a method that exploits certain graphs called (n-1,s)-stars.