Constructions of feebly-one-way families of permutations

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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 {f n} are feebly-one-way of order 2, i.e., the functions f n satisfy the property that, for increasing n, C(f n −1 ) approaches 2 · C(f n) while C(f n) tends to infinity. Both these functions and their corresponding complexities are derived by a method that exploits certain graphs called (n−1,s)-stars.