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Some Fundamental Insights of Computational Complexity Theory

Some Fundamental Insights of Computational Complexity Theory. Avi Wigderson IAS, Princeton, NJ Hebrew University, Jerusalem. ADD. MULT. PRIME. FACTOR. Complexity of Functions. Complexity Classes. Counting Problems Non-DET [Efficient Verification] Efficient Prob. Time

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Some Fundamental Insights of Computational Complexity Theory

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  1. Some FundamentalInsights of ComputationalComplexity Theory Avi Wigderson IAS, Princeton, NJ Hebrew University, Jerusalem

  2. ADD MULT PRIME FACTOR Complexity of Functions

  3. Complexity Classes Counting Problems Non-DET [Efficient Verification] 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

  4. FEASIBLE CANNOT SIMULATE NATURE IS WEAK COMPUTATIONAL COMPUTATIONAL CAN BE DETERMINISTICALLY INCREASED PROBLEM HAS A SECURE PROTOCOL EVERY ZERO PROOFS FOR EVERY THEOREM OF SOME NATURAL CONCEPTS IS IMPOSSIBLE EFFICIENT NO FEASIBLE OF COMPUTATIONAL HARDNESS COMP COMP Axiom: FACTORING is HARD  COMPUTATION RANDOMNESS ENTROPY CRYPTOGRAPHY KNOWLEDGE LEARNING PROOFS FORMAL & RIGOROUS theorems

  5. 3-COL COLORING PLANAR MAPS THM [AH] EVERY PLANAR MAP IS 4-COLORABLE FACT NOT EVERY PLANAR MAP IS 3-COLORABLE

  6. TRIVIAL: 3-COL, FACTOR TRIVIAL: IS TRANSITIVE! THM[C,L,K,S]: 3-COL is NP-Complete THM: IF 3-COL IS EASY THEN FACTOR IS EASY NP – EFFICIENTLY VERIFIABLE PROOFS EFFICIENT REDUCTIONS COMPLETENESS

  7. NP - COMPLETENESS P = NP? Among the most important scientific open problems

  8. EASY MULT FACTOR HARD ALL PARTIES FEASIBLE COMPUTERS • CONTRACT SIGNING • • • PLAYING POKER • • CRYPTOGRAPHY [DH] DIGITAL ENVELOPE [GM] [R] [RSA] • PUBLIC KEY ENCRYPTION • DIGITAL SIGNATURES • THE MILLIONAIRE’S PROBLEM • EVERYTHING!

  9. a a b b     COMPLETE PROBLEM COMPLETE PROBLEM a b OBLIVIOUS COMPUTATION[Y] ALICE BOB f(x,y) || ||      SMALL BOOLEAN CIRCUIT    f(x,y) MANY PLAYERS [GMW] NO CHEATERS!

  10. Alice: Bob: Really?? Convince me! Dr. Alice: Prof. Bob: Really?? Convince me! THM[GMW] 3-Coloring has a ZK-Proof THM[CL] Statement Planar Map M Proof 3-COL of M A Efficient ALG Alice, Bob Alice 1-1 PRIVACY vs. FAULT TOLERANCE Zero Knowledge Interactive Proofs [GMR] • Convincing • Reveal no information THM[GMW] Every theorem has a ZK-Proof Corollary: Fault-tolerant protocols

  11. Statistical test Information Theoretic v(D,D’)=MAX|T(D)-T(D’)| Complexity Theoretic [GM,Y] vc(D,D’)=MAX|T(D)-T(D’)| EffT Computational Indistinguishability DPseudo-Random if THM[BM,Y] p.r. D exits with METRICS ON PROB. DISTRIBUTIONS D probability distribution on {0,1}k Uk uniform distribution

  12. EASY HARD  EFFICIENT A feasible predicate b [B( COMPUTATIONAL ENTROPY  HARDNESS AMPLIFICATION THM[BM,Y]D1=(f(x),b(x)) is pseudorandom THM[BM,Y]Dk=(b(f(k)(x)),...b(f(x)),b(x)) is p.r.

  13. H Hc n n D0- Random x  Un n n+1 D1– Pseudo- Random f(x) b(x) n n+2 f(f(x)) D2– Pseudo- Random b(x) b(f(x)) n n+k f(k+1)(x) b(f(k)(x)) DkPSEUDO-RANDOM b(x) n<<k  C(Factor) || G(x) EFFICIENT [BMY] PSEUDO-RANDOM GENERATORS • PRIVATE KEY CRYPTOGRAPHY • PSEUDO-RANDOM FUNCTION • LEARNING • PROOFS OF HARDNESS • DERANDOMIZING PROBABILISTIC ALGS

  14. C(Factor) HARDNESS vs. RANDOMNESS A efficient probabilistic alg. for h: input z [Y] Det. Simulation: Enumerate all s {0,1}n C(EXP-Time) [NW] a different C(Permanent) pseudo-random generator C(Satisfiability)

  15. PROVE THM THM PROVE OPEN PROBLEMS PROVE “Axiom” PROVE Any Lower Bound PROJECTION REDUCTIONS

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