Pseudorandom Generators and Typically-Correct Derandomization

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Pseudorandom Generators andTypically-Correct
Derandomization

• Jeff Kinne, Dieter van MelkebeekUniversity of
  Wisconsin-Madison
• Ronen Shaltiel
• University of Haifa

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• Just One More Talk

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The Power of Randomness?

• Is randomness more powerful for
• Polynomial-time Algorithms?

• Weaker Derandomization
• IW heuristic
• GW typically-correct

BPP
P
Circuit Testing
PRIMES

• Does BPP P?
• Yes, if pseudorandom generators
• Yes, if circuit lower bounds NW, IW,
• Not without circuit lower bounds KI

Random strings
reject
accept
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Typically-Correct Derandomization

• More efficient derandomizations?
• Weaker (or no) hardness assumptions?
• How to leverage ability to make errors?

• Randomized Algorithm A(x, r) computing L
• Typically-correct B(x) L(x) except for e2n
  xs

• Our Contributions
• New approach based on PRGs
• Simpler proofs, new derandomizations
• Implies circuit lower bounds

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• Previous Approaches to
• Typically-Correct Derandomization

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Goldreich and Wigderson
Randomized Algorithm A(x, r) computing
L Deterministic simulation B(x) A(x, E(x))

• If (1) r lt x and (2) most r correct for all x
• B(x) A(x, x) makes few mistakes
• Make error very small B(x) Majy(A(x, E(x,y)))
• BPP hardness assumption ? PRG ? A satisfies

Subsequent work vMS, Zim, Sha
Set of all r set of all x
perfect r
x
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Shaltiel

• E is 2-O(m)-extractor for x A(x,r) L(x),
  fixed r
• Use PRG to get r lt x
• BPP hardness assumption ? seedless extractor
• Unconditional results for AC0, streaming algs,

• Goal
• PrxA(x,E(x)) L(x) Prx,rA(x,r)
  L(x) 1-?
• Left hand side Sr?0,1m PrxA(x,r)
  L(x)PrxE(x) r A(x,r) L(x)
• Right hand side Sr?0,1m PrxA(x,r)
  L(x)PrxUm r A(x,r) L(x)

Randomized Algorithm A(x, r) computing
L Deterministic simulation B(x) A(x, E(x))

2-m
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• Pseudorandom Generator Approach to
  Typically-Correct Derandomization

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Pseudorandom Generator Approach
Randomized Algorithm A(x, r) computing
L Deterministic simulation B(x) A(x, E(x))

A(G(x))

• E pseudorandom even with seed revealed
• G a seed-extending PRG, G(x) x, E(x)

Goal PrxA(G(x)) L(x) Prx,rA(x, r)
L(x) 1-?
G is pseudorandom against test that checks if
A(x, r) L(x)
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Pseudorandom Generator Approach
Randomized Algorithm A(x, r) computing L B(x)
A(G(x)), G a seed-extending PRG

• Can PRGs be seed-extending?
• Cryptographic No!
• Derandomization Yes! NW,
• Different use of PRG
• B only runs G once, only need poly stretch
• Compare to GW, Sha (PRG extractor)
• PRG is already enough!

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New Results

• New conditional typically-correct
  derandomizations
• New unconditional typically-correct
  derandomizations

Randomized Algorithm A(x, r) computing
L Deterministic simulation B(x) A(x,
NWH(x)) NWH based on hardness of H
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New Conditional Results

• Deterministic polynomial-time simulations of BPP
• Similar conditional results for AM, BPL,

Hardness assumption ?
NW E ? SIZE(2en) 0
GW P is 1/3-hard for SIZESAT(nd) 2ne
Sha P is ½-2-nO(1)-hard for SIZE(nd) 2n/2nO(1)
ours P is 1/poly-hard for SIZE(nd) 2n/poly
mistakes
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New Unconditional Results

• AC0 with few symmetric gates
• A uses o(log2n) sym gates, error ? 1/3
• ? B in AC0sym and B(x)
  L(x) for all but ?n-?(1) fraction of x
• Other settings multi-party communication

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PRGs More General than Sha

• ? PRG approach can prove all of Sha

E is a seedless 2-O(r)-extractor
fordistributions x A(x, r) L(x)
Sha
A(x, E(x)) L(x) for all but ? fraction of x
(x, E(x)) is a 2-O(r)-PRG for tests that check
if A(x,r)L(x)
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• Typically-Correct Derandomizationof BPP
  Implies Circuit Lower Bounds

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Difficulty of Proving Typ-Cor Derand

• KI BPP ? NSUBEXP ? NEXP ? P/poly or PERM
  ? Arith-P/poly
• Typically-correct derandomization of BPP without
  circuit lower bounds?
• No for small error NSUBEXP computes BPP with
  2ne errors
• Large error relativizing techniques and
  arithmetization alone cannot settle

Error rate of GW
Simpler proof for everywhere-correct setting
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Recap

• New seed-extending PRG approach
• simpler proofs, weaker hardness conditions
• unconditional results in some settings!
• BPP setting implies circuit lower bounds, ...

Typically-Correct Derandomization allowed to
make small of mistakes
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• Thanks!
• Full paper and annotated slides
• available from my website