Title: Probabilistic Polynomial-Time Process Calculus for Security Protocol Analysis
1Probabilistic Polynomial-Time Process Calculus
for Security Protocol Analysis
- John Mitchell
- Stanford University
- P. Lincoln, M. Mitchell,
- A. Ramanathan, A. Scedrov, V. Teague
2Computer Security
Goal protection of computer systems and digital
information
Security
- Access control
- OS security
- Network security
- Cryptography
-
Crypto
3Research challenge
- Invent the logic of computer security
- Reasoning principles for systems that use
cryptography and are subject to attack - Analogy
- Effective topos, synthetic domain thy,
- Recursion, recursive domains, collections of
types, form a model of intuitionistic set
theory with additional axioms
4LICS presence at CSFW
Abadi Blanchet Fiore Gordon Gunter
Halpern Jeffrey Kirli Pierce Pavlovic Rusinowitch
Scedrov
Abadi Roscoe
2001
2000
1999
1998
- Check out Crypto, Oakland, CCS,
5Today Protocols and Probability
- Security protocols
- Goals for process calculus
- Specific process calculus
- Probabilistic semantics
- Complexity probabilistic poly time
- Asymptotic equivalence
- Pseudo-random number generators
- Equational properties and challenges
6Protocol Security
- Cryptographic Protocol
- Program distributed over network
- Use cryptography to achieve goal
- Attacker
- Intercept, replace, remember messages
- Guess random numbers, some computation
- Correctness
- Attacker cannot learn protected secret or cause
incorrect conclusion
7IKE 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,
8Simpler 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 fresh response to Alices challenge?
9Important 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(xy) encr(x) encr(y) for
- RSA encrypt(k,msg) msgk mod N
10Standard 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
11Comparison
Hand proofs
?
?
High
Poly-time calculus
Spi-calculus
Athena
Paulson
?
Sophistication of attacks
?
?
?
NRL
?
Bolignano
BAN logic
?
Low
FDR
Murj
?
?
Low
High
Protocol complexity
12Outline
- Security protocols
- Goals for process calculus
- Specific process calculus
- Probabilistic semantics
- Complexity probabilistic poly time
- Asymptotic equivalence
- Pseudo-random number generators
- Equational properties and challenges
13Language 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 CP and CQ
- 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
14Probabilistic 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
15Secrecy 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 plain old RSA if f(i) 2i
16Specification 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
17Nondeterminism vs encryption
- Alice encrypts msg and sends to Bob
- A ? B msg K
- Adversary uses nondeterminism
- Process E0 c?0? c?0? c?0?
- Process E1 c?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
18Semantics
Nondeterministic Semantics
Prove initial results for arbitrary scheduler
19Methodology
- 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
20Outline
- Security protocols
- Goals for process calculus
- Specific process calculus
- Probabilistic semantics
- Complexity probabilistic poly time
- Asymptotic equivalence
- Pseudo-random number generators
- Equational properties and challenges
21Technical 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 ...
22Syntax
- 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
- TT 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
23Probabilistic 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
24Scheduling
- 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
25Example
- Process
- c?rand1? c(x).d?x1? d?2? d(y). e?x1?
- Outer evaluation
- c?1? c(x).d?x1? d?2? d(y). e?x1?
- c?2? c(x).d?x1? d?2? d(y). e?x1?
- Communication
- c?1? c(x).d?x1? d?2? d(y). e?x1?
Each prob ½
Choose according to probabilistic scheduler
26Example (again)
c?rand1? c(x).d?x1? d?2? d(y). e?x1?
Outer Eval
Each with prob 0.5
c?2? c(x).d?x1? d?2? d(y). e?x1?
c?1? c(x).d?x1? d?2? d(y). e?x1?
Comm Step
Choose according to probabilistic scheduler
27Complexity 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 CP halts in time q(nC)
- Minimal evaluation context
- C c(x).d(y) c?20? d?7? e?492?
28Complexity 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/xP
- Substitution bounded by orig length of P
- Size of number m is bounded
- Previous steps preserve occurr of x in P
29Outline
- Security protocols
- Application of process calculus
- Specific process calculus
- Probabilistic semantics
- Complexity probabilistic poly time
- Asymptotic equivalence
- Pseudo-random number generators
- Equational properties and challenges
30How to define process equivalence?
Problem
- Intuition
- Prob CP ? yes - Prob CQ ? yes lt
? - 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 ngt0 indexed by key
length - Asymptotic form of process equivalence
31Probabilistic Observational Equiv
- Asymptotic equivalence within f
- Process, context families Pn ngt0 Qn ngt0
Cn ngt0 - P ?f Q if ? contexts C . ? obs v. ?n0 . ? ngt
n0 . - ProbCnPn ? v - ProbCnQn ?
v lt f(n) - Asymptotically polynomially indistinguishable
- P ? Q if P ?f Q for every polynomial f(n)
1/p(n) - Final defn gives robust equivalence
relation
32Outline
- Security protocols
- Application of process calculus
- Specific process calculus
- Probabilistic semantics
- Complexity probabilistic poly time
- Asymptotic equivalence
- Pseudo-random number generators
- Equational properties and challenges
33Compare 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 nk random bits
- in PUBLIC ?b? end
- P is crypto strong pseudo-random generator
- P ? Q
- Equivalence is asymptotic in security parameter n
34Desired equivalences
- P (Q R) ? (P Q) R
- P Q ? Q P
- P 0 ? P
- P ? Q ? CP ? CQ
- P ? ? c. ( clt1gt c(x).P) x ?FV(P)
- Warning hard to get all of these
35One way to get equivalences
- 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
36Hold for uniform scheduler
- P (Q R) ? (P Q) R
- P Q ? Q P
- P 0 ? P
- P ? Q ? CP ? CQ
Compositionality is important issue in computer
security
37Problem
- Want this equivalence
- P ? ?c. ( clt1gt c(x).P) x ?FV(P)
- Fails for general calculus, general ?
- P d(x).eltxgt
- C ?d.( dlt1gt d(y).elt0gt )
38Comparison
?d.( dlt1gt d(y).elt0gt ?c. ( clt1gt c(x).P) )
left
clt1gt
?d.( dlt1gt d(y).elt0gt d(x).eltxgt )
P
right
clt1gt
left
right
left
elt0gt
elt0gt
elt1gt
elt0gt
elt1gt
Even prioritizing private channels, equivalence
fails
39Paradox
- Two processors connect by network
- Each does private actions
- Unrealistic interaction
- Private coin flip in Beijing does not influence
coin flip in Washington
40Solutions
- Modify scheduler
- Process private channels left-to-right
- Each channel random send-receive pair
- Restrict syntax of protocol, attack
- C P C ?c. ( clt1gt 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
41Current 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,
42Connections with modern crypto
- Cryptosystem consist of three parts
- Key generation
- Encryption
- Decryptions
- Many forms of security
- Semantic security, non-malleability,
chosen-ciphertext security, - Common conditions use prob. games
43Chosen-ciphertext security
- Probabilistic poly-time player A cannot win game
(gt1/2) - A gets public key
- A submits ciphertexts and receives decryptions
- A submits two messages m0, m1 and receives either
? Encr(m0) or ? Encr(m1) at random - A submits ciphertexts ? ? and receives
decryptions - A declares guess g 0 or 1
- Score win if ? Encr(mg), else lose
Deterministic encryption vulnerable to chosen-c
attack
44Simulation security of K,E,D
?
?
pk
m
m
?
?
pk
m
m
pk
K
sk
E
K
D
sk
D
P
plain
cipher
Q
- Algorithms K, E, D indistinguishable from variant
where encryption uses random messages and private
table
Canetti 00 Shoup Pfitzmann-Waidner 00,01
45Goal Chosen-c-secure iff sim-secure
- P ? P1 ? P2 ? ? Q
- Hope to prove using process calculus
- Derive protocol correctness by congruence
- where
- P Game on previous slide
- P1 Same, but quit if some output ? of E seen
before as input to D or output of E - P2 If input ? to D was output by E, use table
instead of algorithm D - Q instead of encrypt, use Encr(0) and table
46Conclusion
- Computer security
- Exacting subject amenable to analysis
- Analysis useful since correctness critical
- Protocols
- Short but complex
- Probabilistic poly-time process calc
- Challenging semantics, proof theory
- Appropriate for game equivalence
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48(No Transcript)
49Chosen-ciphertext security
pk
Key Gen
- A gets public key
- A submits ciphertexts and receives decryptions
- A submits two messages m0, m1 and receives either
? Encr(mi) for i1 or i2 - A submits ciphertexts ? ? and gets decryptions
- A guesses g 0 or 1
- Score win if ? Encr(mg), else lose
sk
Decrypt
m0, m1
choose i0,1
?
mi
Encrypt(mi)
Encrypt
i
50Compositionality
- Property of observational equiv
- A ? B C ? D
- AC ? BD
- similarly for other process forms
51Zero-Knowledge Protocol
P
V
- Witness protection program
- Q(x) iff ? w. P(x,w)
- Prove ?? w. P(x,w) without revealing w
52Identify 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 ?