Time hierarchies for cryptographic function inversion with advice

We prove a time hierarchy theorem for inverting functions computable in a slightly nonuniform polynomial time. In particular, we prove that if there is a strongly one-way function, then for any k and for any polynomial p, there is a function f computable in linear time with one bit of advice such that there is a polynomial-time probabilistic adversary that inverts f with probability 1/p(n) on infinitely many lengths of input, while all probabilistic O(n ^k )-time adversaries with logarithmic advice invert f with probability less than 1/p(n) on almost all lengths of input. We also prove a similar theorem in the worst-case setting, i.e., if P∈-∈NP, then for every l∈>∈k∈ ∈1 (Dtime[nk] ∩Ntime[n]) 1 (Dtime[nl}] ∩ Ntime[n] )1. Bibliography: 21 titles.