Direct Product : Decoding

1
Direct Product Decoding Testing, with
Applications

• Russell Impagliazzo (IAS UCSD)
• Ragesh Jaiswal (Columbia)
• Valentine Kabanets (SFU)
• Avi Wigderson (IAS)

2
Direct-Product (DP) Function
For f U ? R, its k-wise DP function is
fk Uk ? Rk where
fk ( x1, , xk ) ( f(x1), , f(xk) )
3
Applications of Direct Product

• Cryptography
• Derandomization
• Error-Correcting Codes

4
Hardness Amplification
g
Hard function
Harder function
f
Hardness on average
5
(Nonuniform) Hardness on Average
0,1n
0,1n
s
f
0,1n
? 2n
A function f is called d-hard for size s, if
any circuit of size s fails to compute f on at
least d fraction of the inputs.
6
Amplification using Repetition
Intuition If, given a random x, it is hard to
compute f(x), then given k independent random
x1,, xk, it is MUCH HARDER to compute
f(x1),, f(xk).

• Sequential repetition Given an algorithm A,
  ask for ( A(x1), , A(xk) ) ( f(x1),,
  f(xk) )
• 2. Parallel repetition Given an algorithm A,
  ask for A(x1,, xk) ( f(x1),, f(xk) )

Is problem easier if given all k instances at
once ???
7
Direct Product Theorem Yao82, Levin87,
GNW95, Imp95, IW97,

• Suppose
• f is at least d-hard for size s.

• Then
• fk is at least
• (1 - ?)-hard for size
• s spoly(?,?), where ? (1-?)k e-? k
  

0,1nk
0,1n
? 2n
? 2n
8
DP Theorem Constructive Proof Yao82,
Levin87, GNW95, Imp95, IW97,

• Given
• a circuit C (of size s spoly(?,?)) that
• ?-computes fk for
• ? gt (1-?)k e-? k

• Construct
• a circuit C (of size s) that (1-?)-computes f.

0,1nk
0,1n
? 2n
????? ? ??2n
9
Need for List Decoding

• Given a circuit C, there may be at least L
  1/? different functions f1, , fL such that C
  ?-computes fik for each i 1, , L.
• (Partition the inputs into 1/? blocks of
  size ?, and define C to agree with fik on
  block i. )
• Need to allow constructing a list of circuits !

10
(Generic) DP Decoding
X
Repeat O(1/?2) times, and take Majority
r1
X
rk
Need ?(1/?2) correct values f(ri). IW97,
Use non-uniform advice !
C
IJK06 Use C to generate advice !
b1
b
bk
IJKW08 Check C for consistency (on
intersecting k-sets). Trust consistent answers !
if enough bi f(ri), then output b
b
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Our Algorithm IJKW08
Preprocessing Randomly pick a k-set S1
(B1,A)
(with A k/2 ).
x
B1
B2
S1
S2
A
Algo On input x, pick a random S2 (B2, A),
with x2 B2. If C( S1 )A C( S2 )A , then
output C( S2 )x, else re-sample S2
(repeat for lt poly (1/²) iterations ).
Thm With prob ? (²) over (B1, A), Algo (1-
?) computes f.
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Proof Ideas
13
Flowers, cores, petals
Flower determined by S(A,B) Core A Core
values aC(A,B)A Consistent petals
(A,B) C(A,B)A a IJKW08 Flower
analysis
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Structure (Decoding)
Assume C ²-agrees with fk
There are many (²/2) large (²/2) flowers
such that For almost all consistent petals of
the flower, C ¼ fk
Decode g(x) PLURALITY C( S )x
petals S x2 S
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Soundness Amplification
g
More sound proof
Mildly sound proof
f
Proof Oracle (think PCP)
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Graph CSP
Graph G (V, E) Alphabet
S (constant size) Edge
constraints Áe S2 ? 0, 1
(for all e2 E ) Question
f V? S satisfying all Áe ?
PCP Theorem AS, ALMSS For a constant 0lt d lt1,
it is NP-hard to distinguish between
satisfiable graph CSPs and d-unsatisfiable
ones (where every f V? S violates
gt d fraction of edge constraints).
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Graph CSP
Graph G (V, E) Alphabet
S (constant size) Edge
constraints Áe S2 ? 0, 1
(for all e2 E ) Question
f V? S satisfying all Áe ?
PCP Theorem AS, ALMSS For a constant 0lt d lt1,
it is NP-hard to distinguish between
satisfiable graph CSPs and d-unsatisfiable
ones.
queries 2 soundness 1 - d (perfect
completeness)
PCP Proof f V? S Verifier Accept if f
satisfies a random edge
Q2
Q1
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Decreasing soundness by repetition

• sequential repetition proof f V ? S
• soundness 1- d ? (1- d)k
• queries 2k
• parallel repetition proof F Vk ? Sk
• queries 2
• soundness ?

Q1
Q3
Qk-1
Q2
Q4
Qk
19
Decreasing soundness by repetition

• sequential repetition proof f V ? S
• soundness 1- d ? (1- d)k
• queries 2k
• parallel repetition proof F Vk ? Sk
• queries 2
• soundness ?

Q1
Q3
Qk-1
Wish F fk for some f V? S. Then soundness
is (1-d)k !!!
Q2
Q4
Qk
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How ?
proof F Vk? Sk
1. DP-test Make ( 2 ? ) queries to F,
to verify that F fk, for some f V? S.
2. Make 2 parallel-repetition queries
to F verify that all k constraints
are satisfied
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How ?
proof F Vk ? Sk
1. DP-test Make ( 2 ? ) queries to F,
to verify that F fk, for some f V? S.
Wish If DP-Test accepts F with prob
², then F fk . False! The best can hope F
fk on ¼² of inputs. Is it enough ???
Not clear ! Also want to have only 2
queries TOTAL (including parallel repetition
queries) !
22
Direct-Product Testing

• Given an oracle C Uk ? Rk
• Test makes some queries to C, and
• (1) Accepts if C fk.
• (2) Rejects if C is far away from any fk

• (2) If Test accepts C with high
  probability, then C must be close to some
  fk.
• - Want to minimize number of queries to C.
• Want to handle small acceptance probability
• Hope the DP Test will be useful for PCPs

23
DP Testing History

• Given an oracle C Uk ? Rk, is C ¼ gk ?
• queries
  acc. prob.
• Goldreich-Safra 00 20 .99
• Dinur-Reingold 06 2 .99
• Dinur-Goldenberg 08 2 1/ka
• Dinur-Goldenberg 08 2 1/k
• New 3
  exp(-ka)
• New 2
  1/ka
• Derandomization

/
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Consistency tests
25
V-Test GS00,FK00,DR06,DG08
Randomly pick two k-sets S1 (B1,A) and S2
(A,B2)
(with A k1/2 ).
B1
B2
S1
S2
A
Accept if C( S1 )A C( S2 )A
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V-Test Analysis
Theorem DG08 If V-Test accepts with
probability ² gt 1/k?(1) , then there is g U
? R s.t. C ¼ gk on at least ² fraction of
k-sets. When ² lt 1/k, the V-Test does not
work.
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Z-Test
Randomly pick k-sets S1 (B1,A1), S2(A1,B2),
S3(B2,A2)
( A1 A2 m k1/2).
S1
B1
A1
S2
B2
A2
S3
Accept if C( S1 )A1 C( S2 )A1 and C( S2 )B2
C( S3 )B2
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Z-Test Analysis
Theorem (main result) If Z-Test accepts with
prob ² gt exp(-k?(1)), then there is g U? R
s.t. C ¼ gk on at least ² fraction of k-sets.
Also - analyze the V-Test, re-proving DG08
(simpler proof) - analyze
derandomized V-Test and Z-Test
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Proof Ideas
30
Flowers, cores, petals
Flower determined by S(A,B) Core A Core
values aC(A,B)A Consistent petals Cons
(A,B) C(A,B)A a IJKW08 Flower
analysis
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V-Test ) Structure (Testing)
C(A, B1 )E ¼ C(A, B2 )E, with E A
Assume V-Test accepts with prob ²
There are many (²/2) large (²/2) flowers
such that On the petals of the flower, V-test
accepts almost certainly ( 1-poly(²) ).
harmony
E
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V-Test Harmony
For random B1 (E,D1) and B2 (E,D2)
(EA)
Pr B1 2 Cons B2 2 Cons C(A, B1)E ? C(A, B2)E
lt ²4 ltlt ²
Proof Symmetry between A and E (few errors in
AuE ) Chernoff ² ¼ exp(-ka)
D1
B
Intuition Restricted to Cons, an
approx V-Test on E accepts almost surely
Unique Decode!
D2
33
V-Test ) Structure (Testing)
Assume V-Test accepts with prob ²
There are many (²/2) large (²/2) flowers
such that On the petals of the flower, V-test
accepts almost certainly ( 1-poly(²) ).
Main Lemma For g(x) PLURALITY C( S )x
C(S) ¼ gk (S) for almost all (1-²) petals
S.
petals S x2 S
34
Decoding vs. Testing
35
Decoding Testing
Assume V-Test accepts with prob ²
Assume C ²-agrees with fk
There are many large flowers such that For
almost all petals S of the flower, C ¼ fk
There are many large flowers such
that Almost all pairs of intersecting petals
are consistent
Define g(x) PLURALITY C( S )x
petals S x2 S
Conclude C(S) ¼ gk (S) for almost all
petals S of the flower.
Conclude g(x) f(x) for almost inputs x.

36
Back to DP Testing
37
Local DP structure
Field of flowers (Ai,Bi) Each with its own Local
DP function gi Global g?
38
Is there global DP function g ?

• Yes, if ² gt 1/ka DG08 we re-prove
  it
• ( can glue
  together many flowers )
• No, if ² lt 1/k DG08
• But, using Z-Test, we get ² exp( -
  ka) !

39
Counterexample DG
For every x 2 U pick a random gx U ? R For every
k-subset S pick a random x(S) 2 S Define C(S)
gx(S)(S) C(S1)AC(S2)A iff x(S1)x(S2) V-test
passes with high prob ² PrC(S1)AC(S2)A
m/k2 No global g if ² lt 1/k2
40
Back to PCPs
41
2-PCP Amplification History

• f V ? S, F Vk ? Sk VN,
  t log S
• 
  soundness
• Raz 98
  exp( - d3 k/t)
• Holenstein 07
  ( t essential FV)
• Rao 08
  exp( - d2 k )
• 
  ( d2 essential Raz)
• Feige-Kilian 00
  1/ka
• NEW exp ( - d
  k1/2)

Parallel repetition
Projection games
Mix N Match
42
Our PCP Construction
43
A New 2-Query PCP (similar to FK)

• For a regular CSP graph G (V, E), the PCP
  proof
• CE Ek ? (S2)k

Accept if (1) CE (Q1) and CE (Q2) agree on
common vertices, and (2) all edge constraints
are satisfied
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The 2-query PCP Theorem

• Theorem
• If CSP is d unsatisfiable, then no CE is
  accepted with probability gt exp ( - d k1/2).
• 
  
• 
  ( perfect completeness preserved )
• Corollary A 2-query PCP over Sk, of size nk,
  perfect completeness, and soundness exp(- k1/2).

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PCP Analysis

• Edge proof CE lt-gt Vertex proof C
• 1. V-test analysis
• - get Local DP g for the flower Q1
• - Q2 is in the same flower
• - CE (Q2) gk (Q2)
• 2. g violates gt d edges (original soundness)
  .
• 3. Q2 has d violated edges (Chernoff)
  w.r.t. gk
• 4. CE violates some edges of Q2, so Test
  rejects.
• 
  

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Derandomization
47
Derandomized DP Test

• Derandomized DP fk (S), for linear subspaces
  S (as in IJKW08 ) .
• Theorem (Derandomized V-Test)
• If derandomized V-Test accepts C with
  probability ² gt poly(1/k), then there is a
  function g U ? R such that C (S) ¼ gk (S)
  on poly(²) of subspaces S.
• Corollary Polynomial rate testable code.

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Summary

• Direct Product Testing 3 queries
  exponentially small acceptance probability
• Derandomized DP Testing 2 queries
  polynomially small acceptance probability
• 
  ( derandomized V-Test of DG08 )
• PCP 2-Prover parallel k-repetition for
  restricted games, with exponential in k1/2
  decrease in soundness

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Open Questions

• Better dependence on k in our Parallel
  Repetition Theorem exp ( -dk ) ?
• Derandomized 2-Query PCP Obtaining /
  improving
• Moshkovitz-Raz08, Dinur-Harsha09
• using similar techniques.