1. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security1 CSC 774 Advanced Network Security Topic 5.2 Tree-Based Group Diffie Hellman Protocol Acknowledgment: Slides were originally provided by Dr. Yongdae Kim at University of Minnesota.

2. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security2 Membership Operations Formation Member add Member leave Group merge Group partition

3. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security3 Membership Operations Join: a prospective member wants to join Leave: a member wants to (or is forced to) leave Partition: a group is split into smaller groups –Network failure: network event causes disconnectivity –Explicit partition: application decides to split the group Merge: two or more groups merge to form one group –Network fault heal: previously disconnected partitions reconnect –Explicit merge: application decides to merge multiple pre- existing groups into a single group

4. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security4 Tree-Based Group Diffie Hellman Simple: One function is enough to implement it Fault-tolerant: Robust against cascade faults Secure –Contributory –Provable security –Key independence Efficient –d is the height of key tree ( < O(log 2 N)), and N is the number of users –Maximum number of exponentiations per node 3d

5. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security5 Key Tree (General) n4n4 n5n5 gn4n5gn4n5 n6n6 n1n1 n2n2 n3n3 gn2n3gn2n3 gn1gn2n3gn1gn2n3 g g n 1 g n 2 n 3 g n 6 g n 4 n 5 gn6gn4n5gn6gn4n5

6. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security6 Key Tree (n 3 ’s view) gn4gn4 gn5gn5 ggn4n5ggn4n5 gn6gn6 gn1gn1 gn2gn2 n3n3 gn2n3gn2n3 gn1gn2n3gn1gn2n3 GROUP KEY ggn6gn4n5ggn6gn4n5 = g g n 1 g n 2 n 3 g n 6 g n 4 n 5 n3n3 gn2n3gn2n3 gn1gn2n3gn1gn2n3 GROUP KEY Key-path: Set of nodes on the path from member node to root node gn1gn1 gn2gn2 ggn6gn4n5ggn6gn4n5 Co-path: Set of siblings of nodes on the key-path Member knows all keys on the key-path and all blinded keys Any member who knows blinded keys on every nodes and its session random can compute the group key.

7. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security7 Join (n 3 ’s view) n3n3 gn1gn1 gn2gn2 ggn1n2ggn1n2 gn3gn1n2gn3gn1n2 gn4gn4 Tree(n 4 ) n3n3

8. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security8 n3n3 Join (n 3 ’s view) gn1gn1 gn2gn2 ggn1n2ggn1n2 gn3gn1n2gn3gn1n2 gn4gn4 n3n3 gn3n4gn3n4 ggn1n2gn3n4ggn1n2gn3n4

9. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security9 Leave (n 2 ’s view) gn1gn1 n2n2 gn1n2gn1n2 gn3gn3 gn4gn4 ggn1n2gn3n4ggn1n2gn3n4 ggn3n4ggn3n4 n2n2 gn1gn1

10. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security10 Leave (n 2 ’s view) n2n2 gn1n2gn1n2 gn3gn3 gn4gn4 ggn1n2gn3n4ggn1n2gn3n4 ggn3n4ggn3n4 n2n2

11. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security11 Leave (n 2 ’s view) gn3gn3 gn4gn4 ggn3n4ggn3n4 n 2’ g n 2’ g n 3 n 4

12. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security12 Partition (n 5 ’s view) gn4gn4 n5n5 gn4n5gn4n5 gn1gn1 gn3gn3 ggn2n3ggn2n3 ggn1gn2n3ggn1gn2n3 g g n 1 g n 2 n 3 g n 6 g n 4 n 5 gn6gn4n5gn6gn4n5 n6n6 n2n2 gn6gn6 gn2gn2 gn6gn6 gn2gn2 n5n5

13. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security13 Partition (n 5 ’s view) gn4gn4 n5n5 gn4n5gn4n5 gn1gn1 gn3gn3 gn2n3gn2n3

14. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security14 Partition (n 5 ’s view) gn1gn1 gn3gn3 gn4n5gn4n5 gn4gn4 n5n5 n5n5 gn3gn3 n5n5 Change share n 5’ ggn1n3ggn1n3 g n 4 n 5’ g g n 1 n 3 g n 4 n 5’

15. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security15 Partition: Both Sides gn4gn4 n5n5 gn1gn1 gn3gn3 gn6gn6 gn2gn2

16. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security16 Partition: Both sides (N 5 and N 6 ) gn1gn1 gn3gn3 gn2gn2 ggn1n3ggn1n3 n 5’ g n 4 n 5’ g g n 1 n 3 g n 4 n 5’ g n 2 n 6’ n6n6 n2n2 n 6’ gn4gn4

17. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security17 Merge (N 2 ’s view) ggn3n4ggn3n4 gn4gn4 ggn5gn3n4ggn5gn3n4 gn5gn5 gn3gn3 ggn1n2gn5gn3n4ggn1n2gn5gn3n4 ggn6n7ggn6n7 gn7gn7 gn6gn6 gn1n2gn1n2 n2n2 gn1gn1 n2n2 gn1gn1 gn1n2gn1n2 n2n2

18. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security18 Merge (to intermediate node) ggn3n4ggn3n4 gn4gn4 ggn5gn3n4ggn5gn3n4 gn5gn5 gn3gn3 n1n1 n2n2 gn1gn1 gn1n2gn1n2 ggn6n7ggn6n7 gn7gn7 gn6gn6 n2n2 ggn1n2gn6n7ggn1n2gn6n7 gggn1n2gn6n7gn5gn3n4gggn1n2gn6n7gn5gn3n4

19. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security19 Tree Management: do one’s best Join or Merge Policy –Join to leaf or intermediate node, if height of the tree will not increase. –Join to root, if height of the tree increases. Leave or Partition policy –No one can expect who will leave or be partitioned out. –No policy for leave or partition event Successful –Still maintaining logarithmic (height < 2 log 2 N)

20. Computer Science Dr. Peng NingCSC 774 Adv. Net. Security20 Discussion Efficiency –Average number of mod exp: 2 log 2 n –Maximum number of round: log 2 n Robustness is easily provided due to self- stabilization property