Hardness amplification proofs require majority

1
Hardness amplification proofs require majority

• Emanuele Viola
• Columbia University
• Work also done at Harvard and IAS
• Joint work with
• Ronen Shaltiel
• University of Haifa
• May 2008

2
Circuit lower bounds

• Success with restricted circuits
• Furst Saxe Sipser, Ajtai, Yao, Hastad,
  Razborov, Smolensky,
• TheoremRazborov 87 Majority ? AC0Å
• Majority(x) 1 Û å xi gt x/2
• AC0Å Å parity \/ or
• /\ and
• Ø not

Å
constant depth
/\
/\
/\
/\
Å
/\
Ø
Å
Å
Å
V
V
Ø
Input x
3
Natural proofs barrier

• Little progress for general circuit models
• TheoremRazborov Rudich Naor Reingold
• Standard techniques cannot prove lower bounds
  for
• circuit classes that can compute Majority
• We have lower bounds for AC0Å
• because Majority ? AC0Å

4
Average-case hardness

• Definition f 0,1n 0,1 (1/2 - e)-hard for
  class C
• for every M Î C Prxf(x) ? M(x) ³ 1/2 - e
• E.g. C general circuits of size nlog n, AC0Å,
  
• Strong average-case hardness 1/2 e 1/2
  1/nw(1)
• Need for cryptography
• pseudorandom generators Nisan
  Wigderson,
• lower bounds Hajnal Maass Pudlak
  Szegedy Turan,

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Hardness amplificationY,GL,L,BF,BFL,BFNW,I,GNW,
FL,IW,CPS,STV,TV,SU,T,O,V,HVV,GK,IJK,

• Usually black-box, i.e. code-theoretic
• Enc(f) Encoding of (truth-table of) f
• Proof of correctness decoding algorithm in C
• Results hold when C general circuits

Hardness amplification against C
f ? C (lower bound)
Enc(f) (1/2 - e)-hard for C (average-case hardne
ss)
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The problem we study

• Known hardness amplifications fail
• against any class C for which have lower bounds
• ConjectureV. 04 Black-box hardness
  amplification
• against class C Þ Majority Î C

Open f (1/2 - 1/n)-hard for AC0Å ?
Have f ? AC0Å
Hardness amplification against AC0Å
?
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Our results

• TheoremThis work Black-box (non-adaptive)
• (1/2 - e)-hardness amplification against class C
• Þ C computes majority on 1/e bits.
• Tight
• Impagliazzo, Goldwasser Gutfreund Healy
  Kaufman Rothblum

8
Our results Razborov Rudich Naor Reingold
Lose-lose reach of standard techniques
Majority
Power of C
Cannot prove lower bounds RR NR
Cannot prove hardness amplification this work
You can only amplify the hardness you dont know
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Other consequences of our results

• Boolean vs. non-Boolean hardness amplification
• Enc(f)(x) Î 0,1 requires majority
• Enc(f)(x) Î 0,1t does not
  Impagliazzo Jaiswal Kabanets
  Wigderson
• Loss in circuit size Lower bound for size s
• Þ (1/2 - e)-hard for size s?e2/n
• Tight Impagliazzo, Klivans Servedio
• Decoding is more difficult than encoding
• Encoding Parity (Å)
• Decoding Majority

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Outline

• Overview and our results
• Formal statement of our results

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Black-box hardness amplification

• In short " f " h Enc(f) Þ D Î C Dh f
• Rationale f ? C Þ Enc(f) (1/2 - e)-hard for C

0 1 0 1 0 1 0 1 0 ? 1
f
arbitrary
0 1 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 ? 0
Enc(f)
h (1/2 e errors)
0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 ? 0
queries (non-adaptive)
Dh(x) f(x)
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Our results

• Theorem

M Î C computes majority on 1/e bits
Black-box non-adaptive (1/2 - e)-hardness
amplification against C
majority(y)
f(x)
" f, h Enc(f) D Î C Dh f
h
h
x
y 1/e
y
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Proof idea

• (1/2 - e) hardness amplification against C
• Þ D Î C tells Noise rate 1/2 from 1/2 e
• h noise 1/2 Þ Dh ? f
• h Enc(f) Å noise 1/2 e Þ Dh f
• Þ compute majority Ack Madhu Sudan
• Problem D depends on h
• This work Technique to fix D independent of h

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Conclusion

• This work Black-box (non-adaptive)
• hardness amplification against C Þ Majority Î C
• Reach of standard techniques
• This work Razborov Rudich Naor
  Reingold
• Can amplify hardness Û cannot prove lower
  bound
• Open problems
• Adaptivity? (Already can handle special
  cases)
• 1/3-pseudorandom construction Þ majority?