Probabilistic Polynomial-Time Process Calculus for Security Protocol Analysis

Probabilistic Polynomial-Time Process Calculus for Security Protocol Analysis John Mitchell Stanford University P. Lincoln, M. Mitchell, A. Ramanathan, A. Scedrov, V. Teague

Outline • Some discussion of protocols • Goals for process calculus • Specific process calculus • Probabilistic semantics • Complexity – probabilistic poly time • Asymptotic equivalence • Pseudo-random number generators • Equational properties and challenges

Protocol Security • Cryptographic Protocol • Program distributed over network • Use cryptography to achieve goal • Attacker • Intercept, replace, remember messages • Guess random numbers, do computation • Correctness • Attacker cannot learn protected secret or cause incorrect protocol completion

m1 m2 IKE subprotocol from IPSEC A, (ga mod p) B, (gb mod p) , signB(m1,m2) signA(m1,m2) A B Result: A and B share secret gab mod p Analysis involves probability, modular exponentiation, digital signatures, communication networks, …

Simpler: Challenge-Response • Alice wants to know Bob is listening • Send “fresh” number n, Bob returns f(n) • Use encryption to avoid forgery • Protocol • Alice  Bob: { nonce }K • Bob  Alice: { nonce * 5 }K • Can Alice be sure that • Message is from Bob? • Message is in response to one Alice sent?

Important Modeling Decisions • How powerful is the adversary? • Simple replay of previous messages • Decompose, reassemble and resend • Statistical analysis, timing attacks, ... • How much detail in model of crypto? • Assume perfect cryptography • Include algebraic properties • encr(x*y) = encr(x) * encr(y) for RSA encrypt(k,msg) = msgk mod N

Standard analysis methods • Finite-state analysis • Logic based models • Symbolic search of protocol runs • Proofs of correctness in formal logic • Consider probability and complexity • More realistic intruder model • Interaction between protocol and cryptography Easy Hard

Comparison Hand proofs   High Poly-time calculus Spi-calculus Athena  Paulson Sophistication of attacks    NRL  Bolignano BAN logic  Low FDR Murj   Low High Protocol complexity

Outline • Some discussion of protocols Goals for process calculus • Specific process calculus • Probabilistic semantics • Complexity – probabilistic poly time • Asymptotic equivalence • Pseudo-random number generators • Equational properties and challenges

Language Approach [Abadi, Gordon] • Write protocol in process calculus • Express security using observational 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 some idealized version of the protocol Great general idea; application is complicated

Probabilistic Poly-time Analysis • Add probability, complexity • Probabilistic polynomial-time process calc • Protocols use probabilistic primitives • Key generation, nonce, probabilistic encryption, ... • Adversary may be probabilistic • Express protocol and spec in calculus • Security using observational equivalence • Use probabilistic form of process equivalence

Secrecy for Challenge-Response • Protocol P A  B: { i } K B A: { f(i) } K • “Obviously’’ secret protocol Q A  B: { random_number } K B A: { random_number } K • Analysis: P  Q reduces to crypto condition related to non-malleability [Dolev, Dwork, Naor] • Fails for RSA encryption if f(i) = 2i

private channel private channel public channel public channel Specification with Authentication • Protocol P A  B: { random i } K B A: { f(i) } K A  B: “OK” if f(i) received • “Obviously’’ authenticating protocol Q A  B: { random i } K B A: { random j } K i , j A  B: “OK” if private i, j match public msgs

Nondeterminism vs encryption • Alice encrypts msg and sends to Bob • A  B: { msg } K • Adversary uses nondeterminism • Process E0c0 | c0 | … | c0 • Process E1c1 | c1 | … | c1 • Process E c(b1).c(b2)...c(bn).decrypt(b1b2...bn, msg) In reality, at most 2-n chance to guess n-bit key

Probabilistic Semantics 0.5 0.2 0.2 0.3 0.2 0.2 0.5 0.5 0.5 0.3 0.3 0.5 0.2 0.5 0.2 0.5 0.3 0.5 0.3 0.5 Semantics Nondeterministic Semantics Prove initial results for arbitrary scheduler

Methodology • Define general system • Process calculus • Probabilistic semantics • Asymptotic observational equivalence • Apply to protocols • Protocols have specific form • “Attacker” is context of specific form • Induces coarser observational equivalence This talk: general calculus and properties

Outline • Some discussion of protocols • Goals for process calculus Specific process calculus • Probabilistic semantics • Complexity – probabilistic poly time • Asymptotic equivalence • Pseudo-random number generators • Equational properties and challenges

Technical Challenges • Language for prob. poly-time functions • Extend work of Cobham, Cook, Hofmann • Replace nondeterminism with probability • Otherwise adversary is too strong ... • Define probabilistic equivalence • Related to poly-time statistical tests ...

Syntax • Bounded -calculus with integer terms P :: = 0 | cq(|n|) T send up to q(|n|) bits | cq(|n|)(x). P receive | cq(|n|). P private channel | [T=T] P test | P | P parallel composition | ! q(|n|). P bounded replication • Terms may contain symbol n; channel width • and replication bounded by poly in |n|

Probabilistic Semantics • Basic idea • Alternate between terms and processes • Probabilistic evaluation of terms (incl. rand) • Probabilistic scheduling of parallel processes • Two evaluation phases • Outer term evaluation • Evaluate all exposed terms, evaluate tests • Communication • Match send and receive • Probabilistic if multiple send-receive pairs

Scheduling • Outer term evaluation • Evaluate all exposed terms in parallel • Multiply probabilities • Communication • E(P) = set of eligible subprocesses • S(P) = set of schedulable pairs • Prioritize – private communication first • Choose highest-priority communication with uniform (or other) probability

Complexity results • Polynomial time • For each process P, there is a poly q(x) such that • For all n • For all probabilistic schedulers • All minimal evaluation contexts C[ ] eval of C[P] halts in time q(|n|+|C[]|) • Minimal evaluation context • C[ ] = c(x).d(y)…[ ] | c20 | d7 | e492 | …

Complexity: Intuition • Bound on number of communications • Count total number of inputs, multiplying by q(|n|) to account for ! q(|n|). P • Bound on term evaluation • Closed T evaluated in time qT(|n|) • Bound on time for each comm step • Example: cm | c(x).P  [m/x]P • Substitution bounded by orig length of P • Size of number m is bounded • Previous steps preserve # occurr of x in P

Outline • Some discussion of protocols • Application of process calculus • Specific process calculus • Probabilistic semantics • Complexity – probabilistic poly time • Asymptotic equivalence • Pseudo-random number generators • Equational properties and challenges

Problem: How to define process equivalence? • Intuition • | Prob{ C[P] “yes” } - Prob{ C[Q] “yes” } | <  • Difficulty • How do we choose ? • Less than 1/2, 1/4, … ? (not equiv relation) • Vanishingly small ? As a function of what? • Solution • Use security parameter • Protocol is family { Pn } n>0 indexed by key length • Asymptotic form of process equivalence

Probabilistic Observational Equiv • Asymptotic equivalence within f Process, context families { Pn } n>0{ Qn } n>0 { Cn } n>0 P f Q if  contexts C[ ].  obs v. n0 .  n> n0 . | Prob[Cn[Pn] v] - Prob[Cn[Qn] v] | < f(n) • Asymptotically polynomially indistinguishable P  Q if P f Q for every polynomial f(n) = 1/p(n) Final def’n gives robust equivalence relation

Outline • Some discussion of protocols • Application of process calculus • Specific process calculus • Probabilistic semantics • Complexity – probabilistic poly time • Asymptotic equivalence • Pseudo-random number generators • Equational properties and challenges

Compare with standard crypto • Sequence generated from random seed Pn: let b = nk-bit sequence generated from n random bits in PUBLICb end • Truly random sequence Qn: let b = sequence of nkrandom bits in PUBLICb end • P is crypto strong pseudo-random generator P  Q Equivalence is asymptotic in security parameter n

Desired equivalences • P | (Q | R)  (P | Q) | R • P | Q  Q | P • P | 0  P • P  Q  C[P]  C[Q] • P  c. ( c<1> | c(x).P) x FV(P) Warning: hard to get all of these…

r r    ~ ~ ~ How to establish equivalence • Labeled transition system • Allow process to send any output, read any input • Label with numbers “resembling probabilities” • Simulation relation • Relation  on processes • If P Q and P P’, then exists Q’ with Q Q’ and P’ Q’ • Weak form of prob equivalence • But enough to get started …

Hold for uniform scheduler • P | (Q | R)  (P | Q) | R • P | Q  Q | P • P | 0  P • P  Q  C[P]  C[Q]

Problem • Want this equivalence • P c. ( c<1> | c(x).P) x FV(P) • Fails for general calculus, general  • P = d(x).e<x> • C[ ] = d.( d<1> | d(y).e<0> | [ ] )

Comparison • d.(d<1> | d(y).e<0> | c. ( c<1> | c(x).P) ) left c<1> • d.(d<1> | d(y).e<0> | d(x).e<x> ) P right left c<1> right left e<0> e<0> e<1> e<0> e<1> Even prioritizing private channels, equivalence fails

Paradox • Two processors connect by network • Each does private actions • Unrealistic interaction • Private coin flip in Beijing does not influence coin flip in Washington

Solutions • Modify scheduler • Process private channels left-to-right • Each channel: random send-receive pair • Restrict syntax of protocol, attack • C[ P ] = C[ c. ( c<1> | c(x).P) ] for all contexts C[ ] that • do not share private channels • do not bind channel names used in [ ] Modification of scheduler more reasonable for protocols

Current State of Project • Framework for protocol analysis • Determine crypto requirements of protocols • Precise definition of crypto primitives • Probabilistic ptime language • Process framework • Replace nondeterminism with rand • Equivalence based on ptime statistical tests • Methods for establishing equivalence • Develop probabilistic simulation technique • Examples: Diffie-Hellman, Bellare-Rogaway, …

Compositionality • Property of observational equiv A  B C  D A|C  B|D similarly for other process forms

I know a number x with Q(x) Answer these questions Here. Now you’ll believe me. Zero-Knowledge Protocol • Witness protection program • Q(x) iff  w. P(x,w) • Prove  w. P(x,w) without revealing w P V

Identify Friend or Foe • Sequential • One conversation at a time • Concurrent • Base station proves identity concurrently M V A Base S prover verifiers Are concurrent sessions still zero-k ?

Probabilistic Polynomial-Time Process Calculus for Security Protocol Analysis