Towards Interactive Belief, Knowledge & Provability: Possible - PowerPoint PPT Presentation

Computer Laboratory Security Seminar — Cambridge University Towards Interactive Belief, Knowledge & Provability: Possible Application to Zero-Knowledge Proofs ➡ Ph.D. Thesis Chapter 5 Simon Kramer December 18, 2007 Target audience: Cryptographers, Computer Scientists, Logicians, Philosophers

Towards Interactive Belief, Knowledge & Provability Overall Argument 1. Zero-Knowledge proofs have a natural (logical) formulation in terms of modal logic . 2. Modal operators of interactive belief , knowledge , and provability are definable as natural generalisations of their non- interactive counterparts. Simon Kramer, Ecole Polytechnique Paris 2 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Overview 1. Introduction i. Motivation ii. Goal iii.Prerequisites individual knowledge propositional Knowledge spatial implication evidence & Belief, proof & Provability epistemic implication 2. Interactive individual knowledge, proof & Provability 3. Application to Zero-Knowledge proofs 4. Interactive evidence & Belief 5. Conclusion Simon Kramer, Ecole Polytechnique Paris 3 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Motivation How to redefine modern cryptography in terms of modal logic? probabilistic polynomial-time Turing-machines ➡ low- level & operational definitions ( how ) ➡ mentally intractable proofs ➡ Modern cryptography is cryptic. How to generalise non-interactive modal concepts to the interactive setting? [van Benthem] from monologue to dialogue ➡ rational agency (game theory) Simon Kramer, Ecole Polytechnique Paris 4 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Goal To redefine modern cryptography in terms of modal logic ➡ high- level & declarative definitions ( what ) ➡ mentally tractable proofs ➡ Logical cryptology. To define interactive belief, knowledge, and provability ➡ building blocks for rational agency Simon Kramer, Ecole Polytechnique Paris 5 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Prerequisites (1/5) Individual knowledge (knowledge of messages ): • name generation • message reception • message analysis • message synthesis via message synthesis via message analysis ∧ ⊇ Eve k M Eve k k Eve k { | M | } k Eve k k Eve k { | M | } k Eve k M Simon Kramer, Ecole Polytechnique Paris 6 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Prerequisites (2/5) Propositional Knowledge (Knowledge of the truth of propositions) — almost: Simon Kramer, Ecole Polytechnique Paris 7 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Prerequisites (3/5) Spatial implication ( assume — guarantee ): � ǫ · I ( Eve , { | M | } k ) , P � | = Eve k k ⊲ Eve k M ) ) ( ) = Simon Kramer, Ecole Polytechnique Paris 8 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Prerequisites (4/5) Theorem: P a is S4 Provability (other than Artëmov ’ s) & proof: P b ( φ ) := ∃ m ( m proofFor φ ∧ b k m ) m proofFor φ := ∀ ( c : A Adv )( c k m ⊲ K c ( φ )) Theorem: B a is KD4 Belief and evidence: Simon Kramer, Ecole Polytechnique Paris 9 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Introduction Prerequisites (5/5) Epistemic implication ( if — then possibly because ): � · { | | } � | ⊲ � ǫ · I ( Eve , { | M | } k ) · I ( Eve , k ) , P � | = Eve k M ⊇ Eve k k Derivation of individual knowledge { I ( Eve , { | M | } k ) } ǫ · I ( Eve , { | M | } k ) ⊢ ( Eve , { | M | } k ) Eve { I ( Eve , { | M | } k ) } } k ) · I ( Eve , k ) ⊢ { I ( Eve ,k ) } ǫ · I ( Eve , { | M | ( Eve , k ) ǫ · I ( Eve , { | M | } k ) ⊢ { | M | } k Eve Eve { I ( Eve , { | M | } k ) } } k ) · I ( Eve , k ) ⊢ { I ( Eve ,k ) } ǫ · I ( Eve , { | M | ǫ · I ( Eve , { | M | } k ) · I ( Eve , k ) ⊢ { | M | } k k Eve Eve { I ( Eve ,k ) , I ( Eve , { | M | } k ) } ǫ · I ( Eve , { | M | } k ) · I ( Eve , k ) ⊢ M Eve Simon Kramer, Ecole Polytechnique Paris 10 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Interactive individual knowledge, proof & Provability Interactive individual knowledge 2-party interactive proof � � M iProofFor a := M iProofFor ( a,b ) φ ( a,b ) φ ( M, � ) iProofFor c := c k M ∧ M proofFor φ ( a,b ) φ M ′ ⊇ ( a,b ) M ∧ ( M ′ , I ) iProofFor c ( M, ( M ′ , I )) iProofFor c := ( a,b ) φ ( b,a ) φ Simon Kramer, Ecole Polytechnique Paris 11 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Possible Application to Zero- Knowledge Proofs (1/3) 2-party Interactive Provability Zero-Knowledge proofs (definition) Simon Kramer, Ecole Polytechnique Paris 12 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Possible Application to Zero- Knowledge Proofs (2/3) Zero-Knowledge proofs (properties) Simon Kramer, Ecole Polytechnique Paris 13 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Possible Application to Zero- Knowledge Proofs (3/3) Zero-Knowledge proofs (conjecture) Simon Kramer, Ecole Polytechnique Paris 14 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Interactive evidence & Belief 2-party interactive evidence � � M iEvidenceFor a := M iEvidenceFor ( a,b ) φ ( a,b ) φ ( M, � ) iEvidenceFor c := ( a,b ) φ c k M ∧ M evidenceFor φ M ′ ⊇ ( a,b ) M ∧ ( M ′ , I ) iEvidenceFor c ( M, ( M ′ , I )) iEvidenceFor c := ( a,b ) φ ( b,a ) φ 2-party interactive Belief Simon Kramer, Ecole Polytechnique Paris 15 Talk at Cambridge U. on December 18, 2007

Towards Interactive Belief, Knowledge & Provability Conclusion 1. Modern cryptography is cryptic due to its machine-based definitions. 2. This deep-rooted problem must be administered a radical remedy: redefinition . 3. Modal logic is a good candidate remedy. Simon Kramer, Ecole Polytechnique Paris 16 Talk at Cambridge U. on December 18, 2007