Some Fundamental Insights of Computational Complexity Theory

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Some FundamentalInsights of ComputationalComplex
ity Theory

• Avi Wigderson
• IAS, Princeton, NJ
• Hebrew University, Jerusalem

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Complexity of Functions
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Complexity Classes
Counting Problems Non-DET Efficient Verificatio
n Efficient Prob. Time Efficient DET.
Time Memory Efficient ALGS

• Permanent P

• Satisfyability
  NP
• 3-Coloring
• Discrete Log
• Factoring

• Primality testing
  RP
• Verifying polynomial identities

? F E A S I B L E

• Max Flow P
• Linear Programming

• Determinant L
• Graph Connectivity

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COMP
COMP
Axiom FACTORING is HARD ?
COMPUTATION
RANDOMNESS
ENTROPY
CRYPTOGRAPHY
KNOWLEDGE
LEARNING
PROOFS
FORMAL RIGOROUS theorems
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COLORING PLANAR MAPS
THM AH EVERY PLANAR MAP IS 4-COLORABLE
FACT NOT EVERY PLANAR MAP IS 3-COLORABLE
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THM IF 3-COL IS EASY
THEN FACTOR IS EASY
NP EFFICIENTLY VERIFIABLE PROOFS
EFFICIENT REDUCTIONS
COMPLETENESS
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NP - COMPLETENESS
P NP? Among the most important scientific
open problems
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CRYPTOGRAPHY DH
DIGITAL ENVELOPE GM R RSA
PUBLIC KEY ENCRYPTION
DIGITAL SIGNATURES
THE MILLIONAIRES PROBLEM
EVERYTHING!
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OBLIVIOUS COMPUTATION Y
ALICE
BOB
f(x,y)


?
?
?
?
?
SMALL BOOLEAN CIRCUIT
?
?
?
f(x,y)
?MANY PLAYERS GMW
?NO CHEATERS!
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PRIVACY vs. FAULT TOLERANCE
Zero Knowledge Interactive Proofs GMR

• Convincing
• Reveal no information

THMGMW Every theorem has a ZK-Proof
Corollary Fault-tolerant protocols
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METRICS ON PROB. DISTRIBUTIONS
D probability distribution on 0,1k
Uk uniform distribution
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COMPUTATIONAL ENTROPY ?
HARDNESS AMPLIFICATION
THMBM,Y D1(f(x),b(x)) is pseudorandom
THMBM,Y Dk(b(f(k)(x)),...b(f(x)),b(x)) is p.r.
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BMY PSEUDO-RANDOM GENERATORS

• PRIVATE KEY CRYPTOGRAPHY

• PSEUDO-RANDOM FUNCTION

• LEARNING

• PROOFS OF HARDNESS

• DERANDOMIZING PROBABILISTIC ALGS

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HARDNESS vs. RANDOMNESS
A efficient probabilistic alg. for h ? input z
Y Det. Simulation Enumerate all s? 0,1n
C(EXP-Time) NW a different
C(Permanent) pseudo-random generator C(Satisfiab
ility)
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OPEN PROBLEMS
PROVE Axiom
PROVE Any Lower Bound
PROJECTION REDUCTIONS