TR00-022 Authors: Rosario Gennaro, Luca Trevisan

Publication: 9th May 2000 16:16

Downloads: 1313

Keywords:

We present lower bounds on the efficiency 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 as

they match the efficiency of known constructions.

A PRG (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 give lower bounds on the number of invocations to

the oracle by the construction.

If $S$ is the assumed security of the oracle permutation $\pi$

(i.e. no adversary of size $S$ can invert $\pi$ on a fraction

larger than $1/S$ of its inputs) then a PRG (resp. UOWHF) construction that

stretches (resp. compresses) its input by $k$ bits must query $\pi$

in $q=\Omega(k/\log S)$ 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 $P \neq NP$).