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1
CS151
Complexity Theory
Lecture 8
April 21, 2011
April 21, 2011
2
Derandomization
• Goal
: try to simulate BPP in
subexponential time (or better)
• use
PseudoRandom Generator
(PRG):
• often: PRG “good” if it passes (adhoc)
statistical tests
seed
output string
G
t
bits
m
bits
April 21, 2011
3
Derandomization
• adhoc tests not good enough to
prove
BPP has nontrivial simulations
• Our requirements:
– G is
efficiently computable
– “
stretches
”
t
bits into
m
bits
– “
fools
” small circuits: for all circuits C of size at
most
s
:
Pr
y
[C(y) = 1] – Pr
z
[C(G(z)) = 1] ≤
ε
April 21, 2011
4
Simulating
BPP
using PRGs
• Recall: L
BPP
implies exists p.p.t.TM M
x
L
Pr
y
[M(x,y) accepts] ≥ 2/3
x
L
Pr
y
[M(x,y) rejects] ≥ 2/3
• given an input x:
– convert M into circuit C(x, y)
– simplification: pad y so that C = y = m
• hardwire input x to get circuit C
x
Pr
y
[C
x
(y) = 1] ≥ 2/3
(“yes”)
Pr
y
[C
x
(y) = 1] ≤ 1/3
(“no”)
April 21, 2011
5
Simulating
BPP
using PRGs
• Use a PRG G with
– output length
m
– seed length
t
« m
– error
ε
< 1/6
– fooling size
s
= m
• Compute
Pr
z
[C
x
(G(z)) = 1]
exactly
– evaluate
C
x
(G(z))
on every seed
z
{0,1}
t
• running time
(O(m)+(time for G))
2
t
April 21, 2011
6
Simulating
BPP
using PRGs
• knowing
Pr
z
[C
x
(G(z)) = 1]
, can distinguish
between two cases:
0
1/3
1/2
2/3
1
“yes”:
ε
0
1/3
1/2
2/3
1
“no”:
ε
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April 21, 2011
7
BlumMicaliYao PRG
• Initial goal: for all 1 > δ > 0, we will build a
family of PRGs {G
m
} with:
output length
m
fooling size
s
= m
seed length
t =
m
δ
running time
m
c
error
ε
< 1/6
• implies:
BPP
δ>0
TIME(2
nδ
)
(
EXP
• Why? simulation runs in time
O(m+m
c
)(2
m
δ
)
=
O(2
m
2δ
) = O(2
n
2kδ
)
April 21, 2011
8
BlumMicaliYao PRG
• PRGs of this type imply existence of
oneway
functions
– we’ll use widely believed
cryptographic assumptions
Definition
:
One Way Function (OWF)
: function
family f = {f
n
}, f
n
:{0,1}
n
{0,1}
n
– f
n
computable in poly(n) time
– for every family of polysize circuits {C
n
}
Pr
x
[C
n
(f
n
(x))
f
n
1
(f
n
(x))] ≤ ε(n)
– ε(n) = o(n
c
) for all c
April 21, 2011
9
BlumMicaliYao PRG
• believe oneway functions exist
– e.g. integer multiplication, discrete log, RSA
(w/ minor modifications)
Definition
: One Way Permutation: OWF in
which f
n
is 11
– can simplify “
Pr
x
[C
n
(f
n
(x))
f
n
1
(f
n
(x))] ≤ ε(n)
” to
Pr
y
[C
n
(y) = f
n
1
(y)] ≤ ε(n)
April 21, 2011
10
First attempt
• attempt at PRG from OWF f:
– t = m
δ
– y
0
{0,1}
t
– y
i
= f
t
(
y
i1
)
– G(y
0
) =
y
k1
y
k2
y
k3
…y
0
– k = m/t
• computable in time at most
kt
c
< mt
c1
= m
c
April 21, 2011
11
First attempt
• output is “
unpredictable
”:
– no polysize circuit C can output
y
i1
given
y
k1
y
k2
y
k3
…y
i
with nonnegl. success prob.
– if C could, then given
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