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Program Checkers for NP and Black-box separations

Program Checkers for NP and Black-box separations. Mohammad Mahmoody School on Black-Box Impossibilities July 2014. Main Message. Open for 25 years: Do all NP languages have “program checkers”? [ Manuel Blum,  Sampath Kannan : Designing Programs That Check Their Work. STOC 1989 ]

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Program Checkers for NP and Black-box separations

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  1. Program Checkers for NPand Black-box separations Mohammad Mahmoody School on Black-Box Impossibilities July 2014

  2. Main Message • Open for 25 years: Do all NP languages have “program checkers”? [ Manuel Blum, SampathKannan: Designing Programs That Check Their Work. STOC 1989 ] • Message: If NP not checkable  black-box impossibly results follow • Examples: • No NP-hard one-way function [HoMX10] •  No NP-hard hash functions [HaMX10] • No black-box -round ZK for NP from OWPs with negligible soundness error [GWXY] • No black-box 3-round ZK for NP from OWFs with -bit verifier messages [MP12]

  3. Plan • Part I: Short intro to program checkers • Part II: Applications to separations

  4. Plan • Part I: Short intro to program checkers • Part II: Applications to separations

  5. Definition • Program claims to solve language • A program checker gets and as input and runs “safely” : • If P correct : • If Pbuggy:or “I found a bug”. • Both need to hold with “high” probability. • Example: graph isomorphism (or non-isomorphism)

  6. Checking SAT ? • Is SATis checkable? Open since [BK89,FRS89]. • If P(x) says “x satisfiable”  make sure by self-reducibility • What if P(x) = “x not satisfiable”? should still convince the checker… • would be a PCP for Moreover: each query efficiently computable using NP oracle

  7. PCP Two provers • [FRS89]: SAT is checkable iffcoNP is provable with two provers in • Proof: on board! • So: proving coNP with a singleprover in is a stronger tasklets call it : strongprogram checkers for NP. • Known proof systems for coNP require #P –complex provers…

  8. Recalling the Results • NP-hard one-way function  NP checkable • NP-hard hash functions  NP strongly checkable • black-box -round ZK for NP from OWPs with negligible soundness error • No black-box 3-round ZK for NP from OWFs with -bit verifier messages NP has strongprogram checker

  9. Plan • Part I: Short intro to program checkers • Part II: Applications [Just OWF]

  10. Ruling out P  NP Crypto • Open question: Ruling out black-box reductions that prove “NP BPP  OWF exists”(under complexity assumptions) • Potentially easier to rule out stronger primitives (e.g. public-key) Prior works: • [FF 91, BT 04]: non-adaptivereduction • [Brassard 79]: general reduction, but for one-way permutations • [BL 13]: general reduction, but for homomorphic encryption

  11. NP-hard OWF  SAT checkable • Theorem: If R solves NP given any weakly inverting oracle for f There is two prover proof system for coNP with prover complexity • Proof: • Prover 1: emulates the reduction • Prover 2 either: • re-answer one of P1’s answers • or invert P1 rand of R Simulation : (ym,a1),…(ym,am) VER x yi / f(u) ai / u’ P2

  12. Proof Intuition 1) Only one query from P2  it is an “oracle”. 2) If P2 inverts with prob caught with prob 3) P1 should match oracle P2 (or gets caught with prob) 3) Soundness error is high?Use sequential repetition! P1 rand of R Simulation : (ym,a1),…(ym,am) VER x yi / f(u) ai / u’ P2

  13. Direct proof using PCPs • On the board!

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