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
• Natural Proofs Razborov 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
• (i) C Î C computes majority on 1/e bits
• (ii) C Î C makes ³ n/e2 queries
• Generalizes to d (1/2 - e)-hardness
  amplification
• Both tight (i) Impagliazzo, Goldwasser
  Gutfreund Healy Kaufman Rothblum (ii)
  Impagliazzo, Klivans Servedio

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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
• 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
• Proof

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Black-box hardness amplification

• In short " f " h Enc(f) Þ C Î C Ch 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)
Ch(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) C Î C Ch f
h
h
x
y 1/e
y
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Outline

• Overview and our results
• Formal statement of our results
• Proof

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Proof

• Recall Theorem Black-box (non-adaptive)
• (1/2 - e)-hardness amplification against class C
  Þ
• (i) C Î C computes majority on 1/e bits
• (ii) C Î C makes q ³ n/e2 queries
• We show hypot. Þ C Î C tells Noise 1/2 from 1/2
  e
• (D) PrC(N1/2,,N1/2)1 - PrC(N1/2-e,,N1/2-e)
  1 gt0.1
• (i) Ü (D) manipulations Ack Madhu Sudan
• (ii) Ü (D) tigthness of Chernoff bound

q
q
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Warm-up uniform reduction

• Want non-uniform reductions (" f,h C)
• For every f ,h PryEnc(f)(y) ? h(y) lt 1/2-e
• there is circuit C Î C Ch(x) f(x) " x
• Warm-up uniform reductions ( C " f,h )
• There is circuit C Î C
• For every f, h PryEnc(f)(y) ? h(y) lt 1/2-e
• Ch(x) f(x) " x

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Proof in uniform case

• Random F 0,1k 0,1, X Î 0,1k
• Consider C(X) with oracle access to Enc(F)(y) Å
  H(y)
• H(y) N1/2 Þ CEnc(F) Å H(X) CH(X) ? F(X)
  w.h.p.
• C has no information about F
• H(y) N1/2-e Þ CEnc(F) Å H(X) F(X) always
• Enc(F) Å H is (1/2-e)-close to Enc(F)
• To tell z Noise 1/2 from z Noise 1/2 e, z
  q
• Run C(X) answer i-th query yi with Enc(F)(yi) Å
  zi
  Q.e.d.

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Proof outline in non-uniform case

• Non-uniform C depends on F and H (" f,h C)
• Proof outline
• 1) Fix C to C that works for many f,h
• Condition F F C, H H C
• 2) Information-theoretic lemma
• There is good set G Í 0,1n s.t. if all yi
  Î G
• Enc(F) Å H (y1,,yq) Enc(F) Å H
  (y1,,yq)
• Can argue as for uniform case if all yi Î G
• 3) Deal with queries yi not in G

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Fixing C

• Random F 0,1k 0,1, H (x) N1/2 - e
• Enc(F)ÅH is (1/2-e)-close to Enc(F). We have
  ("f,hC)
• With probability 1 over F,H there is C Î C
• C Enc(F) Å H (x) F(x) " x
• Þ there is C Î C with probability 1/C over
  F,H
• C Enc(F) Å H (x) F(x) " x
• Note C all circuits of size poly(k), 1/C
  2-poly(k)

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The information-theoretic lemma

• Lemma
• Let V1,,Vt i.i.d., V1,,Vt V1,,Vt E
• E noticeable Þ there is large good set G Í t
• for every i1,,iq Î G (Vi1,,Viq)
  (Vi1,,Viq)
• Proof E noticeable Þ H(V1,,Vt) large
• Þ H(Vi V1,,Vi -1) large for many i
  (Î G)
• Closeness(Vi1,,Viq),(Vi1,,Viq) ³
  H(Vi1,,Viq)
• ³ H(Viq V1,,Viq -1) H(Vi1 V1
  ,,Vi1-1) large
• 
  Q.e.d.
• Also in Edmonds Rudich Impagliazzo Sgall, Raz

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Applying the lemma

• Vx H(x) Noise 1/2-e
• E H C Enc(F) Å H(x) F(x) " x, PrE
  ³ 1/C
• H H E
• C Enc(F) Å H (x) C Enc(F) Å H (x)
• All queries in G Þ proof for uniform case goes
  thru

0 1 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 ? 0
G
q queries
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Handling bad queries

• Problem C(x) may query bad y Î 0,1n not in G
• Idea Fix bad query. Queries either in G or fixed
  Þ
• proof for uniform case goes thru
• Delicate argument
• Fixing bad query H(y) creates new bad queries
• Instead, fix heavy queries asked by C(x) for
  many xs
• OK because new bad queries are light, affect few
  xs

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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? (OK in special cases V.,
  Gutfreund Rothblum)
• 1/3-pseudorandom construction Þ majority?