TR05-076 Authors: Dima Grigoriev, Edward Hirsch, Konstantin Pervyshev

Publication: 15th July 2005 18:29

Downloads: 2440

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We prove a time hierarchy theorem for inverting functions

computable in polynomial time with one bit of advice.

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

at least 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{n^k} \cap NTime{n})/1 \subsetneq (DTime{n^l} \cap NTime{n})/1.