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Universally Composable Symbolic Analysis of Cryptographic Protocols

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Symbolic Analysis of

Cryptographic Protocols

Ran Canetti and Jonathan Herzog

6 March 2006

The author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the author.

Automated Analysis of

Cryptographic Protocols

Ran Canetti and Jonathan Herzog

6 March 2006

The author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the author.

Overview

- This talk: symbolic analysis can guarantee universally composable (UC) key exchange
- (Paper also includes mutual authentication)
- Symbolic (Dolev-Yao) model: high-level framework
- Messages treated symbolically; adversary extremely limited
- Despite (general) undecidability, proofs can be automated
- Result: symbolic proofs are computationally sound (UC)
- For some protocols
- For strengthened symbolic definition of secrecy
- With UC theorems, suffices to analyze single session
- Implies decidability!

EKA(Na || Nb || B)

EKB(Nb)

Needham-Schroeder-Lowe protocol(Prev: A, B get other’s public encryption keys)

A

B

K

K

Two approaches to analysis

- Standard (computational) approach: reduce attacks to weakness of encryption
- Alternate approach: apply methods of the symbolic model
- Originally proposed by Dolev & Yao (1983)
- Cryptography without: probability, security parameter, etc.
- Messages are parse trees
- Countable symbols for keys (K, K’,…), names (A, B,…) and nonces (N, N’, Na, Nb, …)
- Encryption ( EK(M) ) pairing ( M || N ) are constructors
- Participants send/receive messages
- Output some key-symbol

The symbolic adversary

- Explicitly enumerated powers
- Interact with countable number of participants
- Knowledge of all public values, non-secret keys
- Limited set of re-write rules:

‘Traditional’ symbolic secrecy

- Conventional goal for symbolic secrecy proofs:

“If A or B output K, then no sequence of

interactions/rewrites can result in K”

- Undecidable in general [EG, HT, DLMS] but:
- Decidable with bounds [DLMS, RT]
- Also, general case can be automatically verified in practice
- Demo 1: analysis of both NSLv1, NSLv2
- So what?
- Symbolic model has weak adversary, strong assumptions
- We want computational properties!
- …But can we harness these automated tools?

large class of protocols

‘Soundness’

(need only be done once)

Would like

Simple, automated

What we’d likeSymbolic

protocol

Symbolic

key-exchange

Concrete

protocol

Computational

key-exchange

Some previous work

General area:

- [AR]: soundness for indistinguishability
- Passive adversary
- [MW, BPW]: soundness for general trace properties
- Includes mutual authentication; active adversary
- Many, many others

Key-exchange in particular (independent work):

- [BPW]: (later)
- [CW]: soundness for key-exchange
- Traditional symbolic secrecy implies (weak) computational secrecy

Limitations of ‘traditional’ secrecy

- Big question:

Can ‘traditional’ symbolic secrecy imply standard

computational definitions of secrecy?

- Unfortunately, no
- Counter-example:
- Demo: NSLv2 satisfies traditional secrecy
- Cannot provide real-or-random secrecy in standard models
- Falls prey to the ‘Rackoff’ attack

Achieving soundness

- Soundness requires new symbolic definition of secrecy
- [BPW]: ‘traditional’ secrecy + ‘non-use’
- Thm: new definition implies secrecy (in their framework)
- But: must analyze infinite concurrent sessions and all resulting protocols
- Here: ‘traditional’ secrecy + symbolic real-or-random
- Non-interference property; close to ‘strong secrecy’ [B]
- Thm: new definition equivalent to UC secrecy
- Demonstrably automatable (Demo 2)
- Suffices to consider single session!

(Infinite concurrency results from joint-state UC theorems)

- Implies decidability (forthcoming)

Proof overview (soundness)

Symbolic

key-exchange

- Construct simulator
- Information-theoretic
- Must strengthen notion of UC public-key encryption
- Intermediate step: trace properties(as in [MW,BPW])
- Every activity-trace of UC adversary could also be produced by symbolic adversary
- Rephrase: UC adversary no more powerful than symbolic adversary

Single session UC KE

(ideal crypto)

UC w/ joint state [CR]

(Info-theor.)

Multi-session UC KE

(ideal crypto)

UC theorem

Multi-session KE

(CCA-2 crypto)

Summary & future work

- Result: symbolic proofs are computationally sound (UC)
- For some protocols
- For strengthened symbolic definition of secrecy
- With UC theorems, suffices to analyze single session
- Implies decidability!
- Additional primitives
- Have public-key encryption, signatures [P]
- Would like symmetric encryption, MACs, PRFs…
- Symbolic representation of other goals
- Commitment schemes, ZK, MPC…

Two challenges

- Traditional secrecy is undecidable for:
- Unbounded message sizes [EG, HT] or
- Unbounded number of concurrent sessions

(Decidable when both are bounded) [DLMS]

- Traditional secrecy is unsound
- Cannot imply standard security definitions for computational key exchange
- Example: NSLv2 (Demo)

Prior work: BPW

New symbolic definition

Theory Practice

Implies UC key exchange

(Public-key & symmetric encryption, signatures)

Our work

New symbolic definition:

‘real-or-random’

Theory Practice

Automated verification!

Equiv. to UC key exchange

(Public-key encryption [CH], signatures [P])

UC suffices to examine single protocol run

Decidability?

Demo 3: UC security for NSLv1

Our work: solving the challenges

- Soundness: requires new symbolic definition of secrecy
- Ours: purely symbolic expression of ‘real-or-random’ security
- Result: new symbolic definition equivalent to UC key exchange
- UC theorems: sufficient to examine single protocol in isolation
- Thus, bounded numbers of concurrent sessions
- Automated verification of our new definition is decidable!… Probably

Summary

- Summary:
- Symbolic key-exchange sound in UC model
- Computational crypto can now harness symbolic tools
- Now have the best of both worlds: security and automation!
- Future work

K

Secure key-exchange: UC?

P

P

A

Answer: yes, it matters

- Negative result [CH]: traditional symbolic secrecy does not imply universally composable key exchange

S

K

K

Secure key-exchange: UCP

?

?

P

A

Adversary gets key when output by participants

- Does this matter? (Demo 2)

Secure key-exchange [CW]

P

P

A

- Adversary interacts with participants
- Afterward, receives real key, random key
- Protocol secure if adversary unable to distinguish
- NSLv1, NSLv2 satisfy symbolic def of secrecy
- Therefore, NSLv1, NSLv2 meet this definition as well

large class of protocols

Would like

Main result of talk

(Need only be done

once)

Simple, automated

Analysis strategyDolev-Yao

protocol

Dolev-Yao

key-exchange

Concrete

protocol

UC key-exchange

functionality

{P2, N1, N2}K1

{N2}K2

“Simple” protocols- Concrete protocols that map naturally to Dolev-Yao framework
- Two cryptographic operations:
- Randomness generation
- Encryption/decryption
- (This talk: asymmetric encryption)
- Example: Needham-Schroeder-Lowe

P1

P2

(P1 P2)

(P1 P2)

(P2 P1)

Key P2

Key P1

Key k

Key k

X

Key P2

UC Key-Exchange FunctionalityFKE

(P1 P2)

A

P1

k {0,1}n

(P2 P1)

P2

L

M2

Local output:

Not seen by

adversary

The Dolev-Yao model- Participants, adversary take turns
- Participant turn:

A

P1

P2

Dolev-Yao adversary powers

- Always in Know:
- Randomness generated by adversary
- Private keys generated by adversary
- All public keys

Dolev-Yao key exchange

- Assume that last step of (successful) protocol execution is local output of

(Finished Pi Pj K)

- Key Agreement: If P1 outputs (Finished P1 P2 K)and P2 outputs(Finished P2 P1 K’)thenK = K’.
- Traditional Dolev-Yao secrecy: If Pi outputs

(Finished Pi Pj K), then K can never be in adversary’s set Know

- Not enough!

Goal of the environment

- Recall that the environment Z sees outputs of participants
- Goal: distinguish real protocol from simulation
- In protocol execution, output of participants (session key) related to protocol messages
- In ideal world, output independent of simulated protocol
- If there exists a detectable relationship between session key and protocol messages, environment can distinguish
- Example: last message of protocol is {“confirm”}K where K is session key
- Can decrypt with participant output from real protocol
- Can’t in simulated protocol

Real-or-random (1/3)

- Need: real-or-random property for session keys
- Can think of traditional goal as “computational”
- Need a stronger “decisional” goal
- Expressed in Dolev-Yao framework
- Let be a protocol
- Let r be , except that when participant outputs (Finished Pi Pj Kr),Kr added to Know
- Let f be , except that when any participant outputs (Finished Pi Pj Kr), fresh key Kf added to adversary set Know
- Want: adversary can’t distinguish two protocols

Real-or-random (2/3)

- Attempt 1: Let Traces() be traces adversary can induce on . Then:

Traces(r) = Traces(f)

- Problem: Kf not in any traces of r
- Attempt 2:

Traces(r) = Rename(Traces(f), KfKr)

- Problem: Two different traces may “look” the same
- Example protocol: If participant receives session key, encrypts “yes” under own (secret) key. Otherwise, encrypts “no” instead
- Traces different, but adversary can’t tell

Real-or-random (3/3)

- Observable part of trace: Abadi-Rogaway pattern
- Undecipherable encryptions replaced by “blob”
- Example:

t = {N1, N2}K1, {N2}K2, K1-1

Pattern(t) = {N1, N2}K1, K2, K1-1

- Final condition:

Pattern(Traces(r))

=

Pattern(Rename(Traces(f), KfKr)))

Main results

- Let key-exchange in the Dolev-Yao model be:
- Key agreement
- Traditional Dolev-Yao secrecy of session key
- Real-or-random
- Let be a simple protocol that uses UC asymmetric encryption. Then:

DY() satisfies Dolev-Yao key exchange

iff

UC() securely realizes FKE

Future work

- How to prove Dolev-Yao real-or-random?
- Needed for UC security
- Not previously considered in the Dolev-Yao literature
- Can it be automated?
- Weaker forms of DY real-or-random
- Similar results for symmetric encryption and signatures

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