Computationally Sound Symbolic Protocol Analysis:

Correspondence Theorems

18739A: Foundations of Security and Privacy

Anupam Datta

CMU

Fall 2007-08

Protocol Analysis Techniques

Crypto Protocol Analysis

Formal Models Computational Models

Protocol LogicsModel Checking Inductive Proofs

Dolev-Yao(perfect cryptography)

Random oracleProbabilistic process calculiProbabilistic I/O automata-Correspondence theorems-Computational PCL…

Process Calculi …

Applied -calculusBAN, PCLMurphi, AVISPA Paulson, MSR

Computational and Symbolic Analysis

Cryptography: computational constructions • Define actual algorithms operating on bit strings• For example, RSA is defined as a triple of algorithms

– Key generation: public key is (n,e), private key is d where n=pq for some large primes p,q; ed=1 mod (p-1)(q-1)

– Encryption of m: me mod n– Decryption of c: cd mod n

Formal model: abstract treatment of crypto• Instead of defining cryptographic algorithms, simply

say that they satisfy a certain set of properties– Given ciphertext, obtain plaintext if have exactly the right key– Cannot learn anything from ciphertext without the right key

Goal: Soundness of Formal Analysis

Prove that properties assumed in the formal model are true for some cryptographic constructions

Formal proofs must be sound in following sense:• For any attack in concrete (computational) model…• …there is matching attack in abstract (formal) model• …or else the concrete attack violates computational

security of some cryptographic primitive

If we don’t find an attack in the formal model, then no computational attack exists

– More precisely, probability that a computational attack exists is negligible

Correspondence Theorems

Protocol specification

language

Abstract execution

model

Concreteexecution

model

Securityproperties

Securityproperties

Formal security

Cryptographic security

Soundness theoremRelation

Representative Papers

Abadi, Rogaway. “Reconciling Two Views of Cryptography (The Computational Soundness of Formal Encryption)”. J. Cryptology, 2002.• Symmetric encryption only, passive adversary

Backes, Pfitzmann, Waidner. “Universally Composable Cryptographic Library”. CCS 2003.• Public key encryption, signatures; active adversary

Micciancio, Warinschi. “Soundness of Formal Encryption in the Presence of Active Adversaries”. TCC 2004.• Public-key encryption, active adversary Today

Another Approach

Symbolic Model•PCL Semantics (Meaning of formulas)

Unbounded # concurrent sessions

PCL •Syntax (Properties)•Proof System (Proofs)

Soundness Theorem

(Induction)

High-level proof principles

Cryptographic Model•PCL Semantics (Meaning of formulas)

Polynomial # concurrent sessions

Computational PCL •Syntax ± •Proof System±

Soundness Theorem

(Reduction)

[BPW, MW,…]

Representative Papers

Datta, Derek, Mitchell, Shmatikov, Turuani. “Probabilistic Polynomial-time Semantics for a Protocol Security Logic”. ICALP 2005.• Public-key encryption, active adversary

A. Datta, A. Derek, J. C. Mitchell, B. Warinschi, Computationally Sound Compositional Logic for Key Exchange Protocols, in Proceedings of 19th IEEE Computer Security Foundations Workshop, pp. 321-334, July 2006. • Signatures, key exchange, secure sessions

A. Roy, A. Datta, A. Derek, J. C. Mitchell, Inductive Proofs of Computational Secrecy, in Proceedings of 12th European Symposium On Research In Computer Security , September 2007. • Symmetric and public key encryption; secrecy proof technique

A. Roy, A. Datta, J. C. Mitchell, Formal Proofs of Cryptographic Security of Diffie-Hellman based Protocols, in Proceedings of Symposium on Trustworthy Global Computing, November 2007.• Diffie-Hellman, symmetric and public key encryption, signatures.

Next lecture

Outline

Define what it means for an encryption scheme to be secure against adaptive chosen-ciphertext attack in the multi-user setting

Define a formal language for describing protocols Define concrete trace semantics for protocols

• Actual execution traces of protocols obtained by instantiating nonces with bit strings, etc.

• Traces include actions of the concrete adversary

Show that almost every action of concrete adversary maps to an action of abstract adversary• This will follow from security of encryption scheme

Simple Protocol Language

Pairing and encryption onlyTerm ::= Id | Identity

Key | Public keys onlyNonce |Pair | Ciphertext

Pair ::= (Term, Term)Ciphertext ::= {Term}Key

Can write simple protocols with this syntax• Must describe valid computations of honest parties

– For example, (B receives {X}pk(A), B sends {X}pk(B)) is not

valid because B can’t decrypt {X}pk(A)

Abstract vs. Concrete Execution

A {Na,A}pk(X)

B {Nb,NA}pk(A)

A {Nb}pk(X)

Abstract execution

A bits(m1ex mod nx)

B bits(m2ea mod na)

A bits(m3ex mod nx)

Concrete execution (with RSA)

(ex, nx) is responder’s RSA public key; m1 is a number encoding a concatentation of a random bit string representing nonce Na

and a fixed bit string representing A’s identity

(ea, na) is A’s RSA public key; m2 is a number encoding a concatentation of a random bit string representing nonce Nb

and the bit string extracted from the message received by B

(ex, nx) is responder’s RSA public key; m3 is a number encoding the bit string

extracted from the message received by A

Abstract vs. Concrete Adversary

Fix some set of corrupt users C. Let M be messages sent prior to some point in protocol execution.

• M know(C,M)• If t,t’ know(C,M), then (t,t’) know(C,M)• If (t,t’) know(C,M), then t,t’ know(C,M)

• If t know(C,M), then {t}k know(C,M) for any key k Keys

- Adversary can access any encryption oracle

• If {t}ki know(C,M) and Ai C, then t know(C,M)

- Adversary can decrypt only messages encrypted with public keys of corrupt users

Abstract adversary Concrete adversary

Any polynomial-time algorithm (maybe

probabilistic)

Equivalence of Concrete & Abstract

Need to prove that concrete adversary cannot achieve more than the abstract adversary except with negligible probability• E.g., may guess secret key with negligible

probability

Show that almost any concrete trace is an implementation of some valid abstract trace• Concrete traces represent everything that the

concrete adversary can achieve• If almost any concrete trace can be achieved by

the abstract adversary, it is sufficient to look only at the abstract adversary when doing analysis

Concrete and Abstract Traces

Representation function R maps abstract symbols (nonces, keys, identities) to bit strings• Defines concrete “implementation” of abstract

protocol

Concrete trace is an implementation of an abstract trace if exists a representation function R such that every message in concrete trace is a bit string instantiation of a message in abstract trace• Intuitively, concrete trace is an implementation if it is

created by plugging bit strings in place of abstract symbols in the abstract trace

• Denote implementation relation as

How Can This Fail?

A {Na,A}pk(B)

B ???

Abstract execution

A meb mod nb

B m2eb mod nb

Concrete execution (with RSA)

Suppose encryption scheme is malleable• Can change encrypted message without

decrypting• Plain old RSA is malleable!

Adversary intercepts and squares; B receives m2eb

instead of meb

There is no abstract operation corresponding to

the action of concrete adversary!

…some abstract trace generated by abstract

adversary interacting with abstract encryption and

decryption oracles

is an implementation of…

Every concrete trace generated by concrete adversary interacting with concrete encryption and decryption

oracles...(note that concrete adversary and concrete encryption oracles are

randomized)

There exists an abstract adversary such that…

If the encryption scheme used in the protocol is IND-CCA secure, then for any concrete adversary Ac

ProbRA,Ro(Af: traceset(Af, Of) traceset(Ac(RA),Oc(RO))

1-()

Main Theorem

With overwhelming probability (i.e., dominated by any

polynomial function of security parameter)

Probability is taken over randomness of concrete adversary and encryption oracles

(recall that concrete adversary may be probabilistic and public-key encryption must

be probabilistic)

Proof Outline

1. For any (randomized) concrete adversary, construct the corresponding abstract adversary Af

Abstract “model” of concrete adversary’s behavior Guarantees traceset(Af, Of) traceset(Ac(RA),Oc(RO))

2. Show that every action performed by constructed adversary is a valid action in the abstract model (with overwhelming probability) Abstract adversary is only permitted to decrypt if he

knows the right key (e.g., cannot gain partial info) Prove: constructed adversary doesn’t do anything else

Constructing Abstract Adversary

Concrete adversary Abstract adversary

1. Fix coin tosses (i.e., randomness) of honest participants and concrete adversary

2. This uniquely determines keys and nonces of honest participants Use symbolic constants to give them names in abstract model

3. Every bitstring which is not an honest participant’s key or nonce is considered the adversary’s nonce and given some symbolic name

Must prove that every concrete action can be represented by a valid

abstract action

Reduction to IND-CCA

Prove that every time concrete adversary does something that’s not permitted in abstract model, he breaks IND-CCA security of encryption• This can only happen with negligible probability• Therefore, with overwhelming probability every

concrete action implements some valid abstract action

We’ll prove this by constructing a simulator who runs the concrete adversary in a “box” and breaks IND-CCA exactly when the concrete adversary deviates from the abstract model

Overview of Reduction

Encryption anddecryption oracles(CCA game)

CCA adversary

Concrete adversary Ac

CCA adversary runs concrete adversary as a

subroutine

CCA adversary interacts with encryption and decryption

oracles according to rules of CCA game

To do this, CCA adversary must be able to simulate protocol to Ac (i.e., create an

illusion for Ac that he is interacting with the actual protocol)

CCA adversary wins CCA game exactly when Af does

something illegal in the abstract model

Abstraction Af

What’s Illegal in Abstract Model?

The only way in which constructed abstract adversary Af may violate rules of abstract model is by sending a term with some honest nonce X that he could not have learned by abstract actions• X cannot be learned from messages sent by honest

parties by simple decryption and unpairing rules

Case 1: If Af sends X in plaintext, this means that concrete adversary managed to open encryption• Recall that Af is abstraction of concrete adversary Ac

• If concrete adversary extracts a bitstring (abstracted as nonce X) from under encryption, he wins the CCA game

What if Encryption is Malleable?

Case 2: Af sends {t[X]}k, but neither t[X], nor {t[X]}k was previously sent by honest parties• Adversary managed to take some encryption

containing X and convert it into another encryption containing X

• This is known as malleability. For example, with RSA can convert an encryption of m into encryption of m2

In this case, CCA adversary will win CCA game by using the concrete adversary’s ability to convert one encryption into another• But to do this, CCA adversary must be able to

simulate protocol execution to the concrete adversary

Winning the CCA Game (Simplified)

CCA adversary

users

m0 m1

Left-right selector LR(m0,m1,b)

1. CCA adversary guesses nonce X and picks two values xo and x1

b

…

t(xb)

4. From t(xb) CCA adversary learns whether xo or x1 is inside; outputs bit b correctly

Concrete adversary Ac

2. Every time Ac asks for enck(s(X)), CCA adversary uses oracles to obtain enck(s(xb))

s(xo), s(x1)

enck(s(xb))enck(s(xb))

3. When Ac outputs enck(t(xb)), CCA adversary submits it to decryption oracle

enck(t(xb))enck(t(xb))

Ok, since the ciphertext was not previously created by encryption

oracle

Summary

If the encryption scheme used in the protocol is non-malleable under chosen-ciphertext attack, then the abstract model is sound• If an attack is not discovered in the abstract model,

a concrete attack may exist only with negligible probability

See this paper for a similar result with a formal definition of the operational semantics of the simulator• A. Roy, A. Datta, A. Derek, J. C. Mitchell, Inductive

Trace Properties for Computational Security, invited to appear in Journal of Computer Security.

In more detail…

Abstract execution model and security properties

Concrete execution model and security properties

Relating abstract and concrete security properties

Semantics

• A protocol defines two interactive programs, one for the Initiator and one for the Responder;

• The program of a party A running the NSL protocol with intended responder B:

- Choose random NA, encrypt (A,NA) and send the ciphertext to B;

- Receive a message C; decrypt C with the secret key of A; parse

the plaintext as (B, NA, NB);

- If successful, send the encryption of NB under the public key of B.

(1) I > R : {I,NI}

(2) R > I : {R,NI,NR}

(3) I > R : {NR}

KR

KI

KR

Semantics

• The local state of a party:

- program counter

- an assignment from protocol variables to values, e.g. for the NSL protocol

For the initiator A of a run between parties A, B the initial state: I=A, R=B, KI=KA,KR=KB, NI=NA, NR=?

• The local state of the party changes by receiving/sending messages.

Execution Model

Adv

Environment Env maintains the global state of all parties

• Adversary can create new sessions of the protocol: new(A,B)

new(A,B) A/B Party A A/B Party B

M

N

• Adversary can send messages to any party running the protocol and obtains the message the party would normally output

Abstract Execution Model

AdvA

{C,N}

new(A,B) A/B Party AI=A ; R=B

NI=NA ;NR=?{A,NA

}KB

A/B Party BI=A ; R=B

NI=? ;NR=NB

new(C,A) C/A Party CI=C ; R=A

NI=NC ;NR=?

C/A Party CI=C ; R=A

NI=N ;NR=NA

11 1

1 2

KA

{C,NC}

1

KA

EnvA maintains the global state of all parties;

(1) I > R : {I,NI}

(2) R > I : {R,NI,NR}

(3) I > R : {NR}

KR

KI

KR

• Message are terms from a term algebra

• Adversary is a Dolev-Yao

Abstract Execution Trace

• The interaction between each abstract adversary AdvA and the abstract environment EnvA determines an execution trace

• Denote by Tr(AdvA,EnvA) the trace F0,F1,F2,…,Fn, i.e. the sequence of the global state of the honest parties

F0 F1 Fn…

Abstract Security Properties

• An abstract security property is a predicate PropA on the set of all possible abstract execution traces

• A protocol satisfies abstractly the security property PropA if for all Dolev-Yao adversaries AdvA Tr(AdvA,EnvA) PropA

Concrete Execution Model

AdvC

{c,n}

new(a,b)

{a,nA}PKb

new(c,a)

1

PKa

{c,nc}1

PKa

a/b Party aI=a ; R=b

NI=nA ;NR=?

a/b Party bI=a ; R=b

NI=? ;NR=nB

c/a Party aI=c ; R=a

NI=nC ;NR=?

c/a Party cI=c ; R=a

NI=n ;NR=nA

1 1

1 2

EnvC maintains the global state of all parties(1) I > R : {I,NI}

(2) R > I : {R,NI,NR}

(3) I > R : {NR}

KR

KI

KR

• Messages are bit-strings

• Adversary is any probabilistic polynomial time Turing machine

Concrete Execution Model

AdvC

{c,n}

new(a,b)

{a,nA}PKb

new(c,a)

1

PKa

{c,nc}1

PKa

a/b Party aI=a ; R=b

NI=nA ;NR=?

a/b Party bI=a ; R=b

NI=? ;NR=nB

c/a Party cI=c ; R=a

NI=nC ;NR=?

c/a Party cI=c ; R=a

NI=n ;NR=nA

1 1

1 2

Radv

Renv

EnvC maintains the global state of all parties

• Fixing the randomness of the adversaries and of the parties, determines a unique execution trace

• Denote by Tr(AdvC(Radv),EnvC(Renv)) the trace C0, C1, …, Cn determined by adversary AdvC using randomness Radv and environment EnvC using randomness Renv

Concrete Execution Trace

C0 C1 Cn

Radv

RenvRadv

RenvRadv

Renv

…

Concrete Security Properties

• A concrete security property is a predicate PropC on the set of all possible execution traces

• A protocol satisfies security property PropC if for any probabilistic polynomial-time adversaries AdvC

Tr(AdvC(Radv),EnvC(Renv)) PropC

with overwhelming probability over the random choices Radv and Renv

Relating Abstract/Concrete Security Properties

• Fix PropA and PropC, an abstract and a concrete security property

• PropA < PropC if for all injective functions F, mapping formal values to appropriate bit-strings

F(PropA) PropC

• “Mutual authentication” can be naturally specified in both frameworks, and a relation as the one above holds

Soundness Theorem

THEOREM: Assume formal/computational security properties PropA < PropC and a protocol implemented with an IND-CCA secure encryption scheme. If the abstract execution of the protocol is secure (with respect to PropA), then the concrete execution is secure (with respect to PropC).

Acknowledgements

Thanks to Vitaly Shmatikov and Bogdan Warinschi for providing a number of slides.