An Optimal Certificate Dispersal Algorithm for Mobile Ad Hoc Networks Nagoya Institute of Technology Hua Zheng Shingo Omura Jiro Uchida Koichi Wada

Outline Mobile ad hoc network Mobile ad hoc network Certificate Dispersal Problem Certificate Dispersal Problem Previous Work Previous Work Our New Algorithms Our New Algorithms Some new lower bounds for the problem Some new lower bounds for the problem Conclusions Conclusions Future Work Future Work

Mobile Ad Hoc Network An Ad hoc network is a dynamically changing wireless network that is created by mobile users. (such as PDA, Cell phone) An Ad hoc network is a dynamically changing wireless network that is created by mobile users. (such as PDA, Cell phone) In an ad hoc network mobile users can come and go as their wishes. In an ad hoc network mobile users can come and go as their wishes. Certificate Dispersal System is considered to communicate securely. Certificate Dispersal System is considered to communicate securely.

Public-key & Private-key Each tank holds its public-key and private-key pair for their own. Each tank holds its public-key and private-key pair for their own. private-key public-key

How to encrypt a message A message is encrypted by the public-key. A message is encrypted by the public-key. The encrypted message can only be decrypted by its private-key. The encrypted message can only be decrypted by its private-key.

Public-key dispersal is dangerous public-key This is Mickey ’ s public-key Certificates are needed to obtain the other’s public-key

Certificate When user u trusts in user v, When user u trusts in user v, The certificate from u to v can be issued. The certificate from u to v can be issued. v u private.u private.u

Certificate Authentication

Certificate Graph Nodes ： Mobile users Nodes ： Mobile users Directed Edges ： For any nodes u and v, if there is an issued certificate from u to v, then there is an edge from u to v. Directed Edges ： For any nodes u and v, if there is an issued certificate from u to v, then there is an edge from u to v. vu

Certificate Dispersal Problem Input ： Certificate Graph G Input ： Certificate Graph G Output ： For each node v in G, the set of certificates stored in it s.t. satisfying the following two conditions Output ： For each node v in G, the set of certificates stored in it s.t. satisfying the following two conditions Conditions ： Conditions ： –Connectivity –Completeness

Connectivity For any reachable pair u and v, the certificates on a path which connects them are stored in u and v. For any reachable pair u and v, the certificates on a path which connects them are stored in u and v (2,4) (1,2) (4,5),

Completeness All of the certificates are stored in some node. All of the certificates are stored in some node (2,4) (1,2) (4,5), (2,3) (3,4) (3,1),

Certificate Dispersal Cost The Cost of Certificate Dispersal Algorithm F: The average number of certificates assigned by F to a node in G. The Cost of Certificate Dispersal Algorithm F: The average number of certificates assigned by F to a node in G. Certificate Dispersability Cost of a graph G: The minimum value of the cost of Certificate Dispersal Algorithm on G. Certificate Dispersability Cost of a graph G: The minimum value of the cost of Certificate Dispersal Algorithm on G.

Full Tree Algorithm Full Tree Algorithm –Cost: not more than n-1 Half Tree Algorithm (improved version) Half Tree Algorithm (improved version) –No evaluation in detail Certificate Dispersability Cost Certificate Dispersability Cost –For a directed graph G, c.G  e/n –For a ring G, c.G = n-1 –For a hourglass G, c.G = e/n –For a star graph G, c.G = 2(n-1)/n Eunjin Jung [Certificate Dispersal in Ad hoc Networks] (n: the number of nodes, e: the number of edges)

Graphs we considered Strongly connected graph: Strongly connected graph: –A graph in which for any two distinct nodes, there exists a path between them, is said to be strongly connected. D G =5 –Diameter is the maximum length of a longest distance between any of two nodes.

Graphs we considered Bi-directional graph: Bi-directional graph: –If there is an edge from node u to node v then there exists an edge from v to u, and vice versa u v R G =2 –Radius is the minimum value of the longest length of the shortest path from v to any other nodes, for any node v.

Our Results Graph Dispersability Cost Upper bound Strongly Connected O(D G +e/n) Bi-directional O(R G +e/n) Directed O(pd max +e/n) Lower bound Cube, Mesh, de-Bruijn, k-ary tree  (D G +e/n) D G : diameter of G, R G : radius of G, p: the number of strongly connected components, d max : the maximum diameter of the strongly connected components

Pivot Input ： A strongly connected graph Input ： A strongly connected graph Output ： The set of certificates stored in each node Output ： The set of certificates stored in each node Outline ： Outline ： –Decide a pivot node, –For each node, compute the shortest paths in both directions from the pivot node, –Store all of the certificates on the shortest paths in each direction to that node.

1. Select an arbitrary node as pivot node p p

2. Compute two shortest paths between p and each node in both directions, and store them p (1,2) (2,3), (3,1),

2. Compute two shortest paths between p and each node in both directions, and store them p (2,3), (3,2)

Pivot p (1,2),(2,3),(3,1) (2,3),(3,2) (4,3),(3,4) (5,4),(4,3),(3,6),(6,5) (6,5),(5,4),(4,3),(3,6)

Pivot Pivot satisfies Connectivity Pivot satisfies Connectivity –For any two distinct nodes, there must exist paths via pivot node between them, and we stored all of the certificates on the path to them. Pivot node

CPivot To satisfy Completeness, we store all remaining certificates to pivot node. To satisfy Completeness, we store all remaining certificates to pivot node. Pivot is changed to be a Certificate Dispersal Algorithm, which satisfying both of two conditions. Pivot is changed to be a Certificate Dispersal Algorithm, which satisfying both of two conditions. We name this algorithm as CPivot. We name this algorithm as CPivot.

Evaluation of CPivot Upper bound of the Cost (in the worst case) Upper bound of the Cost (in the worst case) –Strongly connected graph: 2D G +e/n (D G : diameter) Computation time Computation time –O(e)

Evaluation of CPivot More clever choice of pivot node results a better cost. More clever choice of pivot node results a better cost. Upper bound of the Cost (in the worst case) Upper bound of the Cost (in the worst case) –Bi-directional graph: 2R G +e/n (R G : radius) Computation time Computation time –O(ne)

GPivot Input ： A directed graph Input ： A directed graph Output ： The set of certificates stored in each node Output ： The set of certificates stored in each node Note ： A directed graph can be partitioned into strongly connected components, and this partition is unique. Note ： A directed graph can be partitioned into strongly connected components, and this partition is unique.

1. Partition G into strongly connected components

2. Perform Pivot for each component (1,2),(2,3),(3,2) (2,3),(3,1),(1,2) (7,9),(9,7) (6,5),(5,8), (8,6) (8,6),(6,5),(5,8) p

3. Construct a graph in which each node corresponds to each component

C1C1 C2C2 C4C4 C3C3

4. Compute trees rooted at each component C1C1 C2C2 C4C4 C3C3

5. Store all of the certificates on the shortest paths between two pivot nodes C1C1 C2C2 C4C4 C3C3 Store to all of the nodes in C 1

5. Store all of the certificates on the shortest paths between two pivot nodes For all of the other components, do the same operation. For all of the other components, do the same operation. Finally, all unused certificate are stored to an arbitrary node. Finally, all unused certificate are stored to an arbitrary node. This GPivot satisfies Connectivity and Completeness. This GPivot satisfies Connectivity and Completeness.

GPivot (Connectivity) C1C1 C2C2 C4C4 C3C3 1 9 Certificates stored by Pivot Certificates stored in step 5

Evaluation of GPivot Upper bound of the Cost (in the worst case) Upper bound of the Cost (in the worst case) –2d max +(p-1)(2d max +1)+e/n  2pd max +p-1+e/n p:the number of strongly connected components d max :the maximum diameter of the strongly connected components d max :the maximum diameter of the strongly connected components Computation time Computation time –O(p(n+e))

Proof of lower bound G=(V, E), V 1,V 2  V, V 1  V 2 =  G=(V, E), V 1,V 2  V, V 1  V 2 =  Injective Function f: V 1 V 2 Injective Function f: V 1  V 2 P={p(u, f(u)) | u  V 1, u and f(u) are reachable and p(u, f(u)) is a shortest path from u to f(u)} P={p(u, f(u)) | u  V 1, u and f(u) are reachable and p(u, f(u)) is a shortest path from u to f(u)} V1V1 V2V2 f: V 1  V 2

Proof of lower bound Because V 1 and V 2 are disjoint, for satisfying Connectivity, we have to store all of the certificates on the paths in P to the end nodes of each concerned path. Because V 1 and V 2 are disjoint, for satisfying Connectivity, we have to store all of the certificates on the paths in P to the end nodes of each concerned path V1V1V1V1 V2V2V2V2

Proof of lower bound A lower bound depends on one kind of partition pattern and injective function. A lower bound depends on one kind of partition pattern and injective function. P={p(u, f(u)) | u  V 1, u and f(u) are reachable and p(u, f(u)) is a shortest path from u to f(u)} P={p(u, f(u)) | u  V 1, u and f(u) are reachable and p(u, f(u)) is a shortest path from u to f(u)} Lower bound of the Cost Lower bound of the Cost

Proof of lower bound In the case of G is a Bi-directional graph In the case of G is a Bi-directional graph Lower bound of the Cost Lower bound of the Cost

CPivot in Optimal Case Lower bound of the Cost for –Hypercubes –Meshes –Complete k-ary Trees –de-Bruijn graphs The Cost of CPivot equals to these lower bounds. The Cost of CPivot equals to these lower bounds. CPivot is optimal in these cases.

(m,k)-Mesh M k m : M k m : –V( M k m ) ＝ {0, 1, …, k-1} m –E( M k m ) ＝ {(x,y) | x=(a 1,a 2,…,a m ), y=(b 1,b 2,…,b m )  V,  i,  j  i, a j =b j, a i =b i  1} n = k m e = 2m(k m - k m-1 ) M42M42M42M42

|V 1 |=|V 2 |=n/2 |V 1 |=|V 2 |=n/2 Lower bound of Dispersability Cost is k/2 (2,4)-Mesh V1V1V1V1 V2V2V2V2

Lower bound of the Dispersability Cost is km/4 Lower bound of the Dispersability Cost is km/4 Cost of CPivot: 2R G +e/n  km+2m Cost of CPivot: 2R G +e/n  km+2m –e/n=2m-2m/k  2m, R G =km/2 CPivot is an optimal algorithm. CPivot is an optimal algorithm. (m,k)-Mesh

Conclusions We proposed two efficient certificates dispersal algorithms. We proposed two efficient certificates dispersal algorithms. New upper bounds of the certificate dispersability cost for strongly connected graphs and general directed graphs are proved. New upper bounds of the certificate dispersability cost for strongly connected graphs and general directed graphs are proved. Furthermore, our algorithms are optimal for several graph classes. Furthermore, our algorithms are optimal for several graph classes.

Our Results Graph Dispersability Cost Upper bound Strongly Connected O(D G +e/n) Bi-directional O(R G +e/n) Directed O(pd max +e/n) Lower bound Cube, Mesh, de-Bruijn, k-ary tree  (D G +e/n) D G : diameter of G, R G : radius of G, p: the number of strongly connected components, d max : the maximum diameter of the strongly connected components

Future Work The problem that what kind of certificate graphs have lower dispersability cost. The problem that what kind of certificate graphs have lower dispersability cost. To construct some other certificate dispersal algorithms with lower cost for general directed graphs. To construct some other certificate dispersal algorithms with lower cost for general directed graphs. Lower bounds of certificate dispersability cost for other graphs. Lower bounds of certificate dispersability cost for other graphs.