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Reasoning about Concrete Security in Protocol Proofs

This paper explores the quantitative aspects of security in cryptographic protocols, focusing on key refresh rates and the impact of cryptographic primitive vulnerabilities. It aims to provide answers to vital security questions such as how often to refresh keys and how advancements in breaking cryptographic primitives quantitatively affect security. By using a comprehensive framework based on known security properties, the authors analyze protocols employing various cryptographic primitives and establish methodologies for translating security guarantees and managing compositional complexity.

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Reasoning about Concrete Security in Protocol Proofs

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  1. Reasoning about Concrete Security in Protocol Proofs A. Datta, J.Y. Halpern, J.C. Mitchell, R. Pucella, A. Roy

  2. Motivation • We want to answer questions like: • Given a cryptographic protocol and a security property • How frequently should we refresh the keys? • How does any advance in breaking the specific cryptographic primitives used quantitatively affect security? • We base the analysis on the known security properties of the crypto primitives used • A protocol may use a number of different crypto primitives • How do we translate the quantitative guarantees? • How do we handle composition? • Precursor: • Computational PCL [DDMST05,DDMW06,RDDM07,RDM07] • Used to reason about asymptotic security

  3. Security of signatures • Cryptographic Security • Complexity Theoretic • Concrete Existential Unforgeability under Chosen Message Attack Adversary vk Challenger k mi sigk (mi) vk : public verification key k : private signing key m’, sigk (m’) : m’  mi Advantage(Adversary,) = Prob[Adversary succeeds for sec. param. ] • A signature scheme is CMA secure if • Prob-Polytime A. • Advantage (A, ) is a negligible function of 

  4. Security of signatures • Cryptographic Security • Complexity Theoretic • Concrete Existential Unforgeability under Chosen Message Attack Adversary vk Challenger k mi sigk (mi) vk : public verification key k : private signing key m’, sigk (m’) : m’  mi Advantage(Adversary,) = Prob[Adversary succeeds for sec. param. ] • A signature scheme is (t, q, e) - CMA secure if •  t time bounded A making at most q sig queries. • Advantage (A, ) is less than e

  5. A Challenge-Response Protocol m, A n, sigB {m, n, A} A B sigA {m, n, B} • Alice reasons: if Bob is honest, then: • only Bob can generate his signature • if Bob generates a signature of the form sigB{m, n, A}, • he sends it as part of msg2 of the protocol, and • he must have received msg1 from Alice • Alice deduces: Received (B, msg1) Λ Sent (B, msg2)

  6. Computational PCL • Formal Proofs • Syntax, Semantics, Proof System • Proof system for direct reasoning • Verify (X, sigY(m), Y)  Honest (Y)  Sign (Y, m) • No explicit use of probabilities and computational complexity • No explicit arguments about actions of attackers • Semantics capture idea that properties hold with high probability against PPT attackers • Explicit use of probabilities and computational complexity • Probabilistic polynomial time attackers • Soundness proofs one time • Soundness implies result equivalent to security proof by cryptographic reductions

  7. Axiomatizing Security of signatures • Formal Proofs • Syntax, Semantics, Proof System Existential Unforgeability under Chosen Message Attack Adversary vk Challenger k mi sigk (mi) vk : public verification key k : private signing key m’, sigk (m’) : m’  mi Computational PCL:Verify (X, sigY(m), Y)  Honest (Y)  Sign (Y, m) Quantitative PCL:T esig(t,q,) (Verify (X, sigY(m), Y)  Honest (Y)  Sign (Y, m))

  8. Axioms and Proof Rules where,  = esig(t,q,) where, ’ = l()(l()+1)/2 where, Bi are basic steps of the protocol

  9. m, X n, sigY {m, n, X} X Y sigX {m, n, Y}

  10. Previous CPCL Results • Core logic [ICALP05] • Key exchange [CSFW06] • New security definition: key usability • Used by Blanchet et al in CryptoVerif Kerberos proof • Reasoning about computational secrecy [ESORICS07] • Application to Kerberos • Reasoning about Diffie-Hellman [TGC07] • Applications to IKEv2 (standard model) and DH Kerberos (random oracle model)

  11. Logic and Cryptography: Big Picture Protocol security proofs using proof system Axiom in proof system Semantics and soundness theorem Complexity-theoretic crypto definitions (e.g., IND-CCA2 secure encryption) Crypto constructions satisfying definitions (e.g., Cramer-Shoup encryption scheme)

  12. Thanks ! Questions?

  13. Example Property

  14. PCL: Big Picture High-level proof principles • PCL • Syntax (Properties) • Proof System (Proofs) • Computational PCL • Syntax ±  • Proof System±  Soundness Theorem (Induction) Soundness Theorem (Reduction) [BPW, MW,…] • Symbolic Model • PCL Semantics • (Meaning of formulas) • Cryptographic Model • PCL Semantics • (Meaning of formulas) Unbounded # concurrent sessions Polynomial # concurrent sessions

  15. Fundamental Question Conditional first-order logic (Soundness and completeness) [?] ???

  16. Towards QPCL

  17. Protocol language

  18. Conditional implication (OLD) Implication uses conditional probability • [[1  2]] (T,D,) = [[1]] (T,D,)  [[2]] (T’,D,) where T’ = [[1]] (T,D,)

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