Affiliation:
1. University of Haifa, Israel
Abstract
A sampling procedure for a distribution
P
over {0, 1}
ℓ
is a function
C
: {0, 1}
n
→ {0, 1}
ℓ
such that the distribution
C
(
U
n
) (obtained by applying
C
on the uniform distribution
U
n
) is the “desired distribution”
P
. Let
n
>
r
≥ ℓ =
n
Ω(1)
. An ϵ-
nb-PRG
(defined by Dubrov and Ishai [2006]) is a function
G
: {0, 1}
r
→ {0, 1}
n
such that for every
C
: {0, 1}
n
→ {0, 1}
ℓ
in some class of “interesting sampling procedures,”
C
′(
U
r
) =
C
(
G
(
U
r
)) is ϵ-close to
C
(
U
n
) in
statistical distance
.
We construct poly-time computable nb-PRGs with
r
=
O
(ℓ) for poly-size circuits relying on the assumption that there exists β > 0 and a problem
L
in E = DTIME(2
O
(
n
)
) such that for every large enough
n
, nondeterministic circuits of size 2
β
n
that have NP-gates cannot solve
L
on inputs of length
n
. This assumption is a scaled nonuniform analog of (the widely believed) EXP ≠ Σ
2
P
, and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and Ishai have
r
= Ω(ℓ
2
) and are based on very strong cryptographic assumptions or, alternatively, on nonstandard assumptions regarding incompressibility of functions on random inputs. When restricting to poly-size circuits
C
: {0, 1}
n
→ {0, 1}
ℓ
with Shannon entropy
H
(
C
(
U
n
)) ⩽
k
, for ℓ >
k
=
n
Ω(1)
, our nb-PRGs have
r
=
O
(
k
). The nb-PRGs of Dubrov and Ishai use seed length
r
= Ω(
k
2
) and require that the probability distribution of
C
(
U
n
) is efficiently computable.
Our nb-PRGs follow from a notion of “conditional PRGs,” which may be of independent interest. These are PRGs where
G
(
U
r
) remains pseudorandom even when conditioned on a “large” event {
A
(
G
(
U
r
)) = 1}, for an arbitrary poly-size circuit
A
. A related notion was considered by Shaltiel and Umans [2005] in a different setting, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai.
We also give an unconditional construction of poly-time computable nb-PRGs for poly(
n
)-size, depth
d
circuits
C
: {0, 1}
n
→ {0, 1}
ℓ
with
r
=
O
(ℓ · log
d
+
O
(1)
n
). This improves upon the previous work of Dubrov and Ishai that has
r
≥ ℓ
2
. This result follows by adapting a recent PRG construction of Trevisan and Xue [2013] to the case of nb-PRGs. We also show that this PRG can be implemented by a uniform family of constant-depth circuits with slightly increased seed length.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Cited by
5 articles.
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