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Security Analysis of Network Protocols: Logical and Computational Methods John Mitchell Stanford University Logic and Computational Complexity, 2006

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Outline Protocols Some examples, some intuition Symbolic analysis of protocol security Models, results, tools Computational analysis Communicating Turing machines, composability Combining symbolic, computational analysis Some alternate approaches Protocol Composition Logic (PCL) Symbolic and computational semantics

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Many Protocols Authentication Kerberos Key Exchange SSL/TLS handshake, IKE, JFK, IKEv2, Wireless and mobile computing Mobile IP, WEP, i Electronic commerce Contract signing, SET, electronic cash, …

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Mobile IPv6 Architecture IPv6 Mobile Node (MN) Corresponding Node (CN) Home Agent (HA) Direct connection via binding update Authentication is a requirement Early proposals weak

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Supplicant UnAuth/UnAssoc 802.1X Blocked No Key Association i Wireless Authentication MSK EAP/802.1X/RADIUS Authentication 4-Way Handshake Group Key Handshake Data Communication Supplicant Auth/Assoc 802.1X UnBlocked PTK/GTK

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IKE subprotocol from IPSEC A, (g a mod p) B, (g b mod p) Result: A and B share secret g ab mod p AB m1 m2, signB(m1,m2) signA(m1,m2) Analysis involves probability, modular exponentiation, complexity, digital signatures, communication networks

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Needham-Schroeder Protocol { A, NonceA } { NonceA, NonceB } { NonceB} Ka Kb Result: A and B share two private numbers not known to any observer without Ka -1, Kb -1 AB Kb

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Anomaly in Needham-Schroeder AE B { A, Na } { Na, Nb } { Nb } Ke Kb Ka Ke Evil agent E tricks honest A into revealing private key Nb from B. Evil E can then fool B. [Lowe]

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Run of a protocol A B Initiate Respond C D Correct if no security violation in any run Attacker

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Protocol analysis methods Cryptographic reductions Bellare-Rogaway, Shoup, many others UC [Canetti et al], Simulatability [BPW] Prob poly-time process calculus [LMRST…] Symbolic methods Model checking FDR [Lowe, Roscoe, …], Murphi [M, Shmatikov, …], Symbolic search NRL protocol analyzer [Meadows] Theorem proving Isabelle [Paulson …], Specialized logics [BAN, …]

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The Symbolic Model Messages are algebraic expressions Nonce, Encrypt(K,M), Sign(K,M), … Adversary Nondeterministic Observe, store, direct all communication Break messages into parts Encrypt, decrypt, sign only if it has the key Example: K1, Encrypt(K1, hi) K1, Encrypt(K1, hi) hi Send messages derivable from stored parts

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Many formulations Word problems [Dolev-Yao, Dolev-Even-Karp, …] Each protocol step is symbolic function from input message to output message; cancellation law d k e k x = x Rewrite systems [CDLMS] Each protocol step is symbolic function from state and input message to state and output message Logic programming [Meadows NRL Analyzer] Each protocol step can be defined by logical clauses Resolution used to perform reachability search Constraint solving [Amadio-Lugiez, … ] Write set constraints defining messages known at step i Strand space model [MITRE] Partial order (Lamport causality), reasoning methods Process calculus [CSP, Spi-calculus, applied, …) Each protocol step is process that reads, writes on channel Spi-calculus: use for new values, private channels, simulate crypto

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Complexity results (see [Cortier et al]) Bounded # of sessions Unbounded number of sessions Without noncesWith nonces Co-NP completeGeneral: undecidable Bounded msg length: DEXP-time complete Bounded msg length: undecidable Tagged: exptimeTagged: decidable One-copy: DEXP-time complete Ping-pong protocols: Ptime Additional results for variants of basic model (AC, xor, modular exp, …)

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Many protocol case studies Murphi [Shmatikov, He, …] SSL, Contract signing, i, … Meadows NRL tool Participation in IETF, IEEE standards Many important examples Paulson inductive method; Scedrov et al Kerberos, SSL, SET, many more Protocol logic BAN logic and successors (GNY, SvO, …) DDMP …

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Computational model I [Bellare-Rogaway, Shoup, …] Adversary input tape work tape oracle tape AliceBob

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Computational model II [Canetti, …] Turing machine Adversary

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Computational security: encryption Passive adversary Semantic security Chosen ciphertext attacks (CCA1) Adversary can ask for decryption before receiving a challenge ciphertext Chosen ciphertext attacks (CCA2) Adversary can ask for decryption before and after receiving a challenge ciphertext

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Passive Adversary ChallengerAttacker m 0, m 1 E(m i ) guess 0 or 1

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Chosen ciphertext CCA1 ChallengerAttacker m 0, m 1 E(m i ) guess 0 or 1 c D(c)

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Chosen ciphertext CCA2 ChallengerAttacker m 0, m 1 E(m i ) guess 0 or 1 c D(c) c E(m j ) D(c)

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Protocol execution P1P1 P3P3 P4P4 P2P2 output Z Ideal functionality P1P1 P3P3 P4P4 P2P2 F S simulator input Z Protocol security A attacker Slide: R Canetti

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IDEALREAL Trusted party Protocol interaction For every real adversary A there exists an adversary S Universal composability also reactive simulatability [BPW], … see [DKMRS] Slide: Y Lindell

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Symbolic model [NS78,DY84,…] Complexity-theoretic model [GM84,…] Attacker actions - Fixed set of actions, nondeterminism (ABSTRACTION) + Any probabilistic poly- time computation Security properties - Idealized, e.g., secret message = not possessing atomic term representing message (ABSTRACTION) + Fine-grained, e.g., secret message = no partial information about bitstring representation Analysis methods+ Successful array of tools and techniques; automation - Hand-proofs are difficult, error-prone; no automation Can we have best of both worlds?

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Some relevant approaches Simulation framework Backes, Pfitzmann, Waidner Correspondence theorems Micciancio, Warinschi Kapron-Impagliazzo logics Abadi-Rogaway passive equivalence (K2,{01} K3 ), {({101} K2,K5 )} K2, {{K6} K4 } K5 (K2, ), {({101} K2,K5 )} K2, { } K5 (K1, ), {({101} K1,K5 )} K1, { } K5 (K1,{K1} K7 ), {({101} K1,K5 )} K1, {{K6} K7 } K5 Proposed as start of larger plan for computational soundness … [Abadi-Rogaway00, …, Adao-Bana-Scedrov05]

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Symbolic methods compl results Pereira and Quisquater, CSFW 2001, 2004 Studied authenticated group Diffie-Hellman protocols Found symbolic attack in Cliques SA-GDH.2 protocol Proved no protocol of certain type is secure, for >3 participants Micciancio and Panjwani, EUROCRYPT 2004 Lower bound for class of group key establishment protocols using purely Dolev-Yao reasoning Model pseudo-random generators, encryption symbolically Lower bounds is tight; matches a known protocol

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Rest of talk: Protocol composition logic Alices information Protocol Private data Sends and receives Honest Principals, Attacker Send Receive Protocol Private Data Logic now has symbolic and computational semantics

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Example { A, Nonce a } { Nonce a, … } KaKa Kb AB Alice assumes that only Bob has Kb -1 Alice generated Nonce a and knows that some X decrypted first message Since only X knows Kb -1, Alice knows X=Bob

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More subtle example: Bobs view { A, Nonce a } { Nonce a, B, Nonce b } { Nonce b } KaKa Kb AB Bob assumes that Alice follows protocol Since Alice responds to second message, Alice must have sent the first message

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Execution model Protocol Program for each protocol role Initial configuration Set of principals and key Assignment of 1 role to each principal Run x z {x} B ({x} B ) {z} B decr A B C ({z} B ) Position in run

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Formulas true at a position in run Action formulas a ::= Send(P,m) | Receive (P,m) | New(P,t) | Decrypt (P,t) | Verify (P,t) Formulas ::= a | Has(P,t) | Fresh(P,t) | Honest(N) | Contains(t1, t2) | | 1 2 | x | Example After(a,b) = (b a) Notation in papers varies slightly …

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Modal Formulas After actions, condition [ actions ] P where P = princ, role id Before/after assertions [ actions ] P Composition rule [ S ] P [ T ] P [ ST ] P Logic formulated: [DMP,DDMP] Related to: BAN, Floyd-Hoare, CSP/CCS, temporal logic, NPATRL

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Example: Bobs view of NSL Bob knows hes talking to Alice [ receive encrypt( Key(B), A,m ); new n; send encrypt( Key(A), m, B, n ); receive encrypt( Key(B), n ) ] B Honest(A) Csent(A, msg1) Csent(A, msg3) where Csent(A, …) Created(A, …) Sent(A, …) msg1msg3

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Proof System Sample Axioms: Reasoning about possession: [receive m ]A Has(A,m) Has(A, {m,n}) Has(A, m) Has(A, n) Reasoning about crypto primitives: Honest(X) Decrypt(Y, enc(X, {m})) X=Y Honest(X) Verify(Y, sig(X, {m})) m (Send(X, m) Contains(m, sig(X, {m})) Soundness Theorem: Every provable formula is valid in symbolic model

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Modal Formulas After actions, condition [ actions ] P where P = princ, role id Before/after assertions [ actions ] P Composition rule [ S ] P [ T ] P [ ST ] P

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Application DH + CR = ISO Initiator role of DH [ new a ] I Fresh(I, g a ) HasAlone(I, a) Initiator role of CR Fresh(I, m) [send … receive … B… send] Honest(B) ActionsInOrder(…) Combination Substitute g a for m in CR Apply composition rule, persistence Obtain assertion about ISO initiator

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Additional issues Reasoning about honest principals Invariance rule, called honesty rule Preserve invariants under composition If we prove Honest(X) for protocol 1 and compose with protocol 2, is formula still true?

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Composing protocols DH Honest(X) … |- Secrecy |- Authentication |- Secrecy |- Authentication |- Secrecy Authentication [additive] DH CR [nondestructive] ISO Secrecy Authentication = CR Honest(X) …

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PCL Computational PCL PCL Syntax Proof System Symbolic model Semantics Computational PCL Syntax ± Proof System ± Complexity-theoretic model Semantics

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Some general issues Computational PCL Symbolic logic for proving security properties of network protocols using public-key encryption Soundness Theorem: If a property is provable in CPCL, then property holds in computational model with overwhelming asymptotic probability. Benefits Symbolic proofs about computational model Computational reasoning in soundness proof (only!) Different axioms rely on different crypto assumptions

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PCL Computational PCL Syntax, proof rules mostly the same But not sure about propositional connectives… Significant differences Symbolic knowledge Has(X,t) : X can produce t from msgs that have been observed, by symbolic algorithm Computational knowledge Possess(X,t) : can produce t by ppt algorithm Indistinguishable(X,t) : can distinguish from random in ppt More subtle system: some axioms rely on CCA2, some are info-theoretically true, etc.

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Computational Traces Computational trace contains Symbolic actions of honest parties Mapping of symbolic variables to bitstrings Only send-receive actions of the adversary Run of the protocol Set of all possible traces Technicality: we make them equiprobable by explicitly including randomness.

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Complexity-theoretic semantics Given a protocol Q, adversary A T set of all possible traces [[ ]](T) a subset of T that respects in a certain way Intuition: valid when [[ ]](T) is an asymptotically overwhelming subset of T

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Semantics of trace properties Defined in a straight forward way [[Send(X, m)]](T) Contains all traces t such that, t contains a send action by X with the bistring value of the argument corresponding to the bitstring value of m

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Inductive Semantics [[ 1 2 ]] (T) = [[ 1 ]] (T) [[ 2 ]] (T) [[ ]] (T) = T - [[ ]] (T) Implication uses a form of conditional probability [[ 1 2 ]] (T) = [[ 1 ]] (T) [[ 2 ]] (T) where T = [[ 1 ]] (T)

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Semantics of Indist Not a trace property Intuition: Indist(X, m) holds if no algorithm can distinguish m from a random value given Xs view of the run Protocol Attacker CD mView(X) LR(b, m, r) b [[Indist(X, m)]] (T, D, e) = T if | #(t: b=b)-|T|/2 | < e

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Validity of a formula Q |= if adversary A distinguisher D negligible function f n 0 s.t. n > n 0 [[ ]](T,D,f) T(Q,A,n) |[[ ]](T,D,f(n)) | / |T| > 1 – f(n) Fix protocol Q, PPT adversary A Choose value of security parameter n Vary random bits used by all programs Obtain set T=T(Q,A,n) of equi-probable traces

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Proof system Information-theoretic reasoning [new n] X (Y X) Indist(Y, n) Complexity-theoretic reductions Verify(X, m, Y) Honest(X, Y) Y Sign(Y, m) Asymptotic calculations

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Example Axiom Source(Y,u,{m} X ) Decrypts(X, {m} X ) Honest(X,Y) (Z X,Y) Indistinguishable(Z, u) Proof idea: crypto-style reduction Assume axiom not valid: A D negligible f n 0 n > n 0 s.t. [[ ]](T,D,f)|/|T| < 1 –f(n) Construct attacker A that uses A, D to break IND- CCA2 secure encryption scheme Conditional implication essential Parts of proof are similar to [Micciancio, Warinschi]

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Applications of PCL IKE, JFK family key exchange IKEv2 in progress i wireless networking SSL/TLS, 4way handshake, group handshake Kerberos v5 [Cervesato et al] GDOI [Meadows, Pavlovic] Current work Use CPCL to understand computational security of these protocols, reliance on specific crypto properties

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Advantages of Computational PCL High-level reasoning, sound for real crypto Prove properties of protocols without explicit reasoning about probability, asymptotic complexity Composability PCL is designed for protocol composition Identify crypto assumptions needed ISO [DDMW2006] Kerberos V5 [yet unpublished]

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CPCL analysis of Kerberos V5 Kerberos has a staged architecture First stage generates a nonce and sends it encrypted. Second stage uses this nonce as a key to encrypt another nonce. Third stage uses the nonce exchanged in the second stage to encrypt other terms. Our proof system is sufficient to prove the GoodKey-ness of both the nonces. Authentication properties are proved assuming that the encryption scheme provides ciphertext integrity. Modular proofs are made possible by composition theorems.

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Current and Future Work Investigate nature of propositional fragment Non-classical; involves some conditional probability complexity-theoretic reductions connections with probabilistic logics (e.g. Nilsson86) Generalize reasoning about secrecy Extend logic More primitives: signature, hash functions,… Remove current syntactic restrictions on formulas Information-theoretic semantics (thanks to A Scedrov) Only probability; no complexity Other fundamental problems See Kapron-Impagliazzo, etc.

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Conclusion Symbolic model supports useful analysis Tools, case studies, high-level proofs Computational model more correct More accurately reflects realistic attack Two approaches can be combined Several current projects and approaches One example: computational semantics for symbolic protocol logic

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Credits Collaborators M. Backes, A. Datta, A. Derek, N. Durgin, C. He, R. Kuesters, D. Pavlovic, A. Ramanathan, A. Roy, A. Scedrov, V. Shmatikov, M. Sundararajan, V. Teague, M. Turuani, B. Warinschi, … More information Web page on Protocol Composition Logic My web site for related projects not discussed Science is a social process

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