We present l o wer bounds on the e ciency of constructions for Pseudo-Random Generators PRGs and Universal One-Way Hash Functions UOWHFs based on black-box access to one-way permutations. Our lower bounds are tight a s t h e y m a t c h the e ciency of known constructions. A P R G resp. UOWHF construction based on black-box access is a machine that is given oracle access to a permutation. Whenever the permutation is hard to invert, the construction is hard to break. In this paper we g i v e l o

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... wer bounds on the number of invocations to the oracle by the construction. If S is the assumed security of the oracle permutation i.e. no adversary of size S can invert on a fraction larger than 1=S of its inputs then a PRG r e s p . U O WHF construction that stretches resp. compresses its input by k bits must query in q = k=log S points. This matches known constructions. Our results are given in an extension of the Impagliazzo-Rudich model. That is, we prove that a proof of the existence of PRG resp. UOWHF black-box constructions that beat our lower bound would imply a proof of the unconditional existence of such construction which would also imply P 6 = N P .