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Analysis of Security Protocols (III)

Analysis of Security Protocols (III). John C. Mitchell Stanford University. Analyzing Security Protocols. Non-formal approaches (can be useful, but no tools…) Some crypto-based proofs [Bellare, Rogaway] BAN and related logics Axiomatic semantics of protocol steps

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Analysis of Security Protocols (III)

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  1. Analysis of Security Protocols (III) John C. Mitchell Stanford University

  2. Analyzing Security Protocols • Non-formal approaches (can be useful, but no tools…) • Some crypto-based proofs [Bellare, Rogaway] • BAN and related logics • Axiomatic semantics of protocol steps • Methods based on operational semantics • Intruder model derived from Dolev-Yao • Protocol gives rise to set of traces • Perfect encryption • Possible to include known algebraic properties

  3. Example projects and tools • Prove protocol correct • Paulson’s “Inductive method”, others in HOL, PVS, etc. • Bolignano -- Abstraction methods • MITRE -- Strand spaces • Process calculus approach: Abadi-Gordon spi-calculus • Search using symbolic representation of states • Meadows: NRL Analyzer, Millen: Interrogator • Exhaustive finite-state analysis • FDR, based on CSP [Lowe, Roscoe, Schneider, …] • Mur -- specialized input language • Clarke et al. -- state search with axiomatic intruder model

  4. Explicit Intruder Method Informal Protocol Description Formal Protocol Intruder Model Analysis Tool Gee whiz. Looks OK to me.

  5. Formal Logic (      ) Proof search (Horn clause) Multiset rewriting Finite Automata Process Calculus A notation for inf-state systems • Define protocol, intruder in minimal framework • Disadvantage: need to introduce new notation

  6. Protocol Notation • Non-deterministic infinite-state systems • Facts F ::= P(t1, …, tn) t ::= x | c | f(t1, …, tn) • States { F1, …, Fn } • Multiset of facts • Includes network messages, private state • Intruder will see messages, not private state Multi-sorted first-order atomic formulas

  7. State Transitions • Transition • F1, …, Fkx1 … xm. G1, … , Gn • What this means • If F1, …, Fkin state , then a next state ’ has • Facts F1, …, Fk removed • G1, … , Gnadded, with x1 … xm replaced by new symbols • Other facts in state  carry over to ’ • Free variables in rule universally quantified • Pattern matching in F1, …, Fkcan invert functions

  8. Finite-State Example a • Predicates:State, Input • Function: • Constants:q0, q1, q2, q3, a, b, nil • Transitions:State(q0), Input(a  x)  State(q1), Input(x) State(q0), Input(b  x)  State(q2), Input(x) ... a q1 b a q0 b q3 b a b q2 b

  9. Existential Quantification • Natural-deduction proof rule [y/x] ( elim) x.   • Summary: for proof from x., choose new symbol and proceed from [y/x] y not free in any other hypothesis

  10. Turing Machine • Predicates • Current(state,cell)-- current state, tape pos. • Contents(cell, symbol)-- contents of tape cell • Adjacent(cell, cell)-- keep cells in order • Constants • q0, q1, q2, …-- finite set of states • c0, ceot-- initial tape cells • “0”, “1”, “b”-- tape symbols

  11. Turing Machine (II) • Transitions • Adjacent(c0, ceot) • Adjacent(c, ceot)  c’. Adjacent(c,c’), Adjacent(c’,ceot) • Current(qi,c), Contents(c,“0”), Adjacent(c,c’)  Current(qk,c’), Contents(c,“1”), Adjacent(c,c’) • Current(qi,c), Contents(c,“1”), Adjacent(c’,c)  Current(qk,c’),Contents(c,“0”), Adjacent(c,c’) ... c eot ... c c’ eot infinite linear tape sample move right sample move left

  12. Simplified Needham-Schroeder • Predicates Ai, Bi, Ni --Alice, Bob, Network in state i • Transitions x. A1(x) A1(x)  N1(x), A2(x) N1(x)  y. B1(x,y) B1(x,y)  N2(x,y), B2(x,y) A2(x), N2(x,y)  A3(x,y) A3(x,y)  N3(y), A4(x,y) B2(x,y), N3(y)  B3(x,y) picture next slide A  B: {na, A}Kb B  A: {na, nb}Ka A  B: {nb}Kb • Authentication A4(x,y)  B3(x,y’)  y=y’

  13. A  B: {na, A}Kb B  A: {na, nb}Ka A  B: {nb}Kb Sample Trace A1(na) x. A1(x) A1(x)  A2(x), N1(x) N1(x)  y. B1(x,y) B1(x,y)  N2(x,y), B2(x,y) A2(x), N2(x,y)  A3(x,y) A3(x,y)  N3(y), A4(x,y) B2(x,y), N3(y)  B3(x,y) N1(na) A2(na) B1(na, nb) A2(na) N2(na, nb) B2(na, nb) A2(na) B2(na, nb) A3(na, nb) N3(nb) B2(na, nb) A4(na, nb) B3(na, nb) A4(na, nb)

  14. Common Intruder Model • Derived from Dolev-Yao model [1989] • Adversary is nondeterministic process • Adversary can • Block network traffic • Read any message, decompose into parts • Decrypt if key is known to adversary • Insert new message from data it has observed • Adversary cannot • Gain partial knowledge • Guess part of a key • Perform statistical tests, …

  15. Formalize Intruder Model • Intercept and remember messages N1(x)  M(x) N2(x,y)  M(x), M(y) N3(x)  M(x) • Send messages from “known” data M(x)  N1(x), M(x) M(x), M(y)  N2(x,y), M(x), M(y) M(x)  N3(x), M(x) • Generate new data as needed x. M(x) Highly nondeterministic, same for any protocol

  16. Attack on Simplified Protocol A1(na) x. A1(x) A1(x)  A2(x), N1(x) N1(x)  M(x) x. M(x) M(x)  N1(x), M(x) N1(x)  y. B1(x,y) N1(na) A2(na) A2(na) M(na) A2(na) M(na), M(na’) N1(na’) A2(na) M(na), M(na’) A2(na) M(na), M(na’) B1(na’, nb) Continue “man-in-the-middle” to violate specification

  17. Modeling Perfect Encryption • Encryption functions and keys • For public-key encryption • two key sorts: e_key, d_key • predicate Key_pair(e_key, d_key) • Functions enc :e_key  msg -> msg dec :d_key  msg -> msg (implicit in pattern-matching) • Properties of this model • Encrypt, decrypt only with appropriate keys • Only produce enc(key, msg) from key and msg (!!!) • This is not true for some encryption functions

  18. Steps in public-key protocol • Bob generates key pair and publishes • e_keyu. d_keyv. Bob1(u,v) • Bob1(u,v)NAnnounce(u),Bob2(u,v) • Alice sends encrypted message to Bob • Alice1(e,d,x), NAnnounce(e’)Alice2(e,d,x,e’) • Alice2(e,d,x,e’) N1(enc(e’,x,e)), Alice3(u,v,x,w) • Bob decrypts • Bob1(u,v), N1(enc(u, x,y))z. Bob1(u,v,x,y,z)

  19. Intruder Encryption Capabilities • Intruder can encrypt with encryption key • Me(k), Mdata(x)  Ni(enc(k,x)), Me(k), Mdata(x) • Intruder can decrypt with decryption key • Nj(enc(k,x)),Key_pair(k,k’), Md(k’),  Mdata(x), ... • Add to previous intruder model Assumes sorts data, e_key, d_key with typed predicates Mdata(data), Me(e_key), Md(d_key)

  20. Intruder: power and limitations • Can find some attacks • Needham-Schroeder by exhaustive search • Other attacks are outside model • Interaction between protocol and encryption • Some protocols cannot be modeled • Probabilistic protocols • Steps that require specific property of encryption • Possible to prove erroneous protocol correct • Requires property that crypto does not provide

  21. Optimize Protocol + Intruder • Adversary receives all messages; no net • Replace • Alicei(x,y)  Nj(x), Alicek(x,y) • Nj(x)  M(x) • M(z)  Nj(z), M(z) • Nj(x), Bobi(w)  Bobj(w,y) • By • Alicei(x,y)  M(x), Alicek(x,y) • M(z), Bobi(w)  Bobj(w,y) Alice’s message can go to Bob or M. M can replay or send different msg All messages go directly to M. M can forward or send different msg

  22. Analysis of Protocol+Intruder • Prove properties of protocols • Unbounded # of participants, message space • Prove that system satisfies specification • Paulson, etc: prove invariant holds at all reachable states • Spi-calculus: prove protocol equivalent ideal protocol • Symbolic search with pruning • Search backward from error • Prune search by proving forward invariants • Exhaustive finite-state methods • Approximate infinite-state system by finite one • Search all states, perhaps with optimizations

  23. Example description languages • First- or Higher-order Logic • Define set of traces, prove protocol correct • Horn-clause Logic x… (A1A2 …  B) • Symbolic search methods • Process calculus • FDR model checker based on CSP • Spi-calculus proof methods based on pi-calculus • Additional formalisms • CAPSL protocol description language [Millen] • Mur language for finite-state systems

  24. Paulson’s Inductive Method • Define set TR of traces of protocol+intruder • Similar to traces in unifying formalism • Transition F1, …, Fkx1 … xm. G1, … , Gn gives one way of extending trace • Auxiliary functions mapping traces to sets • Analz(trace) = data visible to intruder • Synth(trace) = messages intruder can synthesize • Definitions and proofs use induction • Similar inductive arguments for many protocols

  25. Symbolic Search Methods • Examples: NRL Protocol Analyzer, Interrogator • Main idea • Write protocol as set of Horn clauses • Transition F1, …, Fkx1 … xm. G1, … , Gn can be Skolemized and translated to Prolog clauses • Search back from possible error for contradiction • This is usual Prolog refutation procedure • Important pruning technique • Prove invariants by forward reasoning • Use these to avoid searching unreachable states

  26. Process Calculus Description • Protocol defined by set of processes • Each process gives one step of one principal • Can derive by translation from unifying notation • F1, …, Fkx1 … xm. G1, … , Gn is one process • Replace predicates by port names • Replace pattern-matching by explicit destructuring • In pi-calculus, use  in place of  • Example • B1(x,y) N2(x,y), B2(x,y) • b1(p). let x=fst(p) and y=snd(p) in n2x,y| b2 x,y end

  27. Spi-Calculus [AG97, ...] • Write protocol in process calculus • Express security using observ. equivalence • Standard relation from programming language theory P  Q iff for all contexts C[ ], same observations about C[P] and C[Q] • Context (environment) represents adversary • Use proof rules for  to prove security • Protocol is secure if no adversary can distinguish it from an idealized version of the protocol

  28. Finite-state methods • Two sources of infinite behavior • Many instances of participants, multiple runs • Message space or data space may be infinite • Finite approximation • Transitions: F1, …, Fkx1 … xm. G1, … , Gn choose fixed number of Skolem constants • Terms: restrict repeated functions f(f(f(f(x)))) • Can express finite-state protocol + intruder in • CSP : FDR-based model checking projects • Other notations: Mur project, Clarke et al., ...

  29. Security Protocols in Mur • Standard “benchmark” protocols • Needham-Schroeder, TMN, … • Kerberos • Study of Secure Sockets Layer (SSL) • Versions 2.0 and 3.0 of handshake protocol • Include protocol resumption • Discovered all known or suspected attacks • Recent work on tool optimization [Shmatikov, Stern, ...]

  30. Malleability [Dolev,Dwork,Naor] • Idealized assumption • If intruder produces Network(enc(k,x)) then either • Network(enc(k,x))  M (enc(k,x)) (replay) • M(k), M(x)  M (enc(k,x))(knows parts) • Not true for RSA • encrypt(k,msg) = msgk mod N • property encr(x*y) = encr(x) * encr(y) • Model • Network(enc(k,x))  M (…) ...  Network (enc(k,c*x)) Can send encrypted message without “knowing” message Finite state ?

  31. Authentication and Secrecy for Handshake Protocols • How many protocols are there to verify? • Average length 7 steps • Data fields per message 5 fields • Distinct ways to fill a field  50 entries • Number of possible combinations 1750 protocols • Research directions • Get the monkeys and typewriters going • Easier description and specification, faster tools • Improved analysis of timestamps, ... • Interaction between protocol and crypto primitives

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