1. Recent developments in group key exchange
Mike Burmester
Information Security Summer School 2005
Florida State University

2. Outline 1. Secure Communication 2. Key Distribution
the Diffie-Hellman protocol
variants, attacks
authentication
conference protocols
3. Public Key Certificates
trust-graphs
hierarchical vs horizontal structures
security
4. Conclusion

3. 1. Secure Communication Security issues authenticity privacy
message
Sender
(Alice)
Receiver
(Bob)
Adversary
Security issues
authenticity
privacy
denial of service, etc.

4. Symmetric keys (privacy)
Bob
Alice
plaintext
ciphertext
plaintext
private channel E D SK SK
Security issue
How to distribute the secret key SK

5. Public Keys (privacy) E D Alice Bob Authentication channel f
plaintext
ciphertext
plaintext E D
SKB
PKB
Authentication channel f
Security issues
It should be hard to compute SKB from PKB
How do we distribute PKB

6. Public Keys (digital signatures)
Bob
Alice
m, sigSKA m a m or r S V
SKA
Authentication channel
PKA f
Security issues
It should be hard to compute SKA from PKA
How to distribute PKA

7. 2. Key Exchange protocols
the Diffie-Hellman protocol
Zp = {0,1,…,p-1}, p prime, g a generator of Zp*
Alice’s Public Key gsa: 0 < sa< p-1, private key sa
Bob’s Public Key gsb: < sa< p-1, private key sb
gsa mod p
Alice
Bob
gsb mod p
Key Exchanged: SK = gsasb
mod p

8. Security Freshness of keys It should be hard to compute SK from PK.
If the same key is used many times then the
security of the system may be undermined.

9. What if 3 or more parties want to sha re a common secret key?
Use DH to get: SKAB , SKBD ,
SKBE , SKAC , SKCF .
K/SKAB
K/SKAC B C
.A selects the secret key K
at random from Zp*.
K/SKBD
.A sends K/SKAB to
B and K/SKAC to C. D E F
4. B gets K from K /SKAB and sends K/SKAC to D, etc.

10. – contributory schemes
Group Key Exchange
– contributory schemes U2 U3 U1
Round 1: Use DH
Ui broadcasts zi = gri Un
Un-1

11. Group Key Exchange U2 U3 K23 Ki2 … Round 1: U1 Kn-1n Un Knn-1 … Un-1
Each Ui computes
the DH key:
Ki = gri ri+1 U1
Kn-1n Un
Knn-1 …
Un-1

12. Group Key Exchange U2 U3 K23 Ki2 … Round 1: end U1 But how???? Kn-1n
K = K1K2 … Kn
Where Ki = Ki,i+1
But how???? U1
Kn-1n Un
Knn-1 …
Un-1

13. Group Key Exchange U2 U3 K2 Ki … Round 2: Ui broadcasts U1
xi = Ki/Ki-1 U1 Kn Un
Kn-1 …
Un-1

14. Group Key Exchange U2 U3 K2 Ki … U1 Kn Round 2: Each Ui computes
the key:
K = Ki-1n zin-1 zi+1n-2 … zi-2
= Ki-1n (Ki/Ki-1)n-1(Ki+1/Ki)n-2… (Ki-1/Ki-2) Un
Kn-1 …
Un-1

15. Authentication 1 How does Alice know that the “shared”
secret key has been distributed to all the parties in the conference?

16. Group Key Exchange – authentication
Each Ui authenticates (digitally signs) its
randomness ri
its zi and xi
and after checking them authenticates
the string:
{Ui}|| {ri} || {zi} || {xi}

17. Authentication 2
How can Alice be certain which key is Bob’s public key?
1. They may have met earlier and exchanged public
keys.
2. They may have mutual friends who know their
public keys:
Alice Carol Bob, or
Alice Carol Bob
Case 1 establishes an a priori trust relationship
Case 2 establishes an induced trust relationship

18. 3. Public Key Certificates
Who is who?
PK CERTIFICATE
The public key of Bob is: …..
Signed by a Certifying Authority
A PK Certificate establishes authenticity and
provides a means by which a public key can
be stored in partially insecure repositories, or
transmitted over insecure channels.

19. Trust-graphs A B C D E F Certificates can be used to
Model the confidence of
a network in its public
keys by a directed
trust-graph, with
vertices the entities
and edges the
certificates.
CAB
CAC B C
CBD
CBE
CCF D E F

20. Trust-graphs A priori confidence: Induced confidence:
This is corroborated by the certificates.
Induced confidence:
This is established by trust-paths that link the entities in the trust-graph.

21. A hierarchical infrastructure
RCA
CA2
CA1 U4 U3 U1 U2
The public key of U4 is certified by the trust-path:
RCA CA U4

22. Security issues A hacker can penetrate a CA or its
computer system and forge certificates or get certificates for unauthorized users.

23. Threats 1. Whom should we trust (and for what)? 2. Which Bob is it?
3. Organizational (insider) attacks
4. Computer system threats:
How secure is the computer system
of the Certifying Authority?
of Bob?

24. PGP: an unstructured approach
Pretty Good Privacy is a freeware electronic
mail system that uses an unstructured
authentication framework.
Users are free to decide whom they trust.
PGP does not specify any specific structure
for the trust-graph and for this reason is
quite vulnerable.
A A An B

25. A horizontal approach: multiple connectivity
If the trust-graph is (2k+1)-connected then
there are 2k+1 vertex disjoint trust-paths
which connect any two of its vertices

26. A 3-connected trust-graphB

27. Combining horizontal and hierarchical structures

28. Security
A secure authentication infrastructure must be, reliable, robust and survivable.
Reliability deals with faults that occur in a
random manner, and is achieved by replication.
Robustness deals with maliciously induced
faults.

29. Survivability deals with the destruction of
parts of the infrastructure.
The destruction may affect the entities
(e.g. the CA’s) as well as stored data, and
may be malicious.
For survivability, the remaining entities should be able to recover enough of the infrastructure to guarantee secure communication.

30. Survivability Reconstruction of a corrupted trust-graph
Adversary
faulty
U U U Un A
Entity A asks all its neighbors for a list of their neighbors, the neighbors of their neighbors, etc

31. Survivability Problem Some of the neighbors are under the control of
the Adversary and may send fake certificates,
relating to other entities, real or bogus.
Is it possible to reconstruct a sufficiently good
approximation of the trust-graph?

32. Survivability Answer Yes, provided that there is a bound on the number
of penetrated or destroyed cites, and that the
trust-graph is sufficiently connected.

33. Reconstructing a corrupted
trust-graph
The reconstruction involves several stages.
Round Robin flooding
a Halting routine
a Clean-up routine

34. Conclusion Secure key exchange can be achieved in
several ways by using cryptographic
mechanisms.
Clearly there is a trade off between the security
requirements and the complexity.

35. Conclusion If the public keys are authenticated via single
trust paths then the system is vulnerable to any
penetration.
By having several vertex disjoint authentication
paths linking the entities we get robustness
against penetration and survivability.