Leave Protocol. Tree T5 Tree T5 <0,0> <0,0> <1,0> <1,1> <1,0> <1,1> <2,0> <2,1> <2,2> <2,3> M3 <3,6> <3,7> <2,0> <2,1> <2,2> <2,3> M1 M2 M1 M2 M4 M5 M4 M5 Sponsor
Leave Protocol. ^ Such as Join protocol, we start with n members and assume that member Md leaves the group. The sponsor in this case is the rightmost leaf node of the subtree rooted at leaving member’s sibling node. First, if the number of leaving member’s sibling node is two, each member updates its key tree by deleting the leaf node corresponding to Md. Then the former sibling of Md is updated to replace Md’s parent node. Otherwise each member only deleting the leaf node corresponding to Md. The sponsor generates a new key share, computes all [key, bkey] pairs on the key-path up to the root, and broadcasts the new set of bkey. This allows all members to compute the new group key. In Fig. 3, if member M7 leaves the group, every remaining member deletes < 1, 2 > and < 2, 6 >. After updating the tree, the sponsor (M10) picks a new share K<2,8>, recomputes X<0,0>, X<0,0>, XX<0,0> and BK<1,2>, and broadcasts the updated tree T10 with BK1∗0. Upon receiving the broadcast message, all members compute the group key. Note that M7 cannot compute the group key, though he knows all the bkeys, because his share is no longer a part of the group key.
Leave Protocol. Assume that a member ML wishes to leave a n-member group. First ML initiates the leave protocol by sending a leave request. When the other group members receive the request, they independently determine the sponsor node, which is defined as in [1] to be the right-most leaf node of the subtree rooted at the leaving member’s sibling node. The leave protocol works as shown in Algorithm 2. Algorithm 2 Leave Protocol in AFTD
Leave Protocol. Once again, we start with members and assume that member leaves the group. The sponsor in this case is the rightmost leaf node of the subtree rooted at the leaving member’s sibling node. First off, as shown in Figure 4, each member updates its key tree by deleting the leaf node corresponding to . The former sibling of is promoted to replace ’s parent node. The sponsor generates a new key share, computes all pairs on the key-path up to the root, and broadcasts the new set of bkeys. This allows all members to compute the new group key. Looking at the setting in Figure 5, if member leaves the group, every remaining member deletes and . After updating the tree, the sponsor ( ) picks a new share , recomputes and , and broadcasts the updated tree with . Upon receiving the broadcast message, all members compute the group key. Note that cannot compute the group key, though it knows all the bkeys, because its share is no longer part of the group key. One round and one message are required to complete a leave protocol. The number of modular exponentiation depends on the location of the leaving member and tree structure. Its upper bound is if all pairs on the key-path of the deepest node need to be recomputed. When either left or right subtree has single node and it is the sponsor (i.e. for example, its sibling leaves the group), 3 modular exponentiations are required (two by the sponsor and one by all other members).
Leave Protocol. Consider the group of n members. Assume that member Md ,d leaves the group by broadcasting the leave message that contains its ip address. Upon receiving it every other member in a group updates its tree structure using binary search tree deletion rules. There are three cases for updating the tree and selecting a sponsor node, which is defined as follows If the leaving node o Does not have any children, Take the sibling as the sponsor node if it is present ,otherwise parent node is the sponsor node Then just remove the node from the tree o Have only one child Replace the leaving node by the child node Then the same node is taken as a sponsor node o Have two children The sibling or parent node of the smallest valued node in the right sub tree if the leaving node is replaced by the node with smallest value from its right sub tree Or the sibling or parent node of the largest valued node in the left sub tree if the leaving node is replaced by the node with largest value from its left sub tree After the selection of sponsor node, Figure 3.7 after a Node 7 Leaves the Group from Figure 3.3
Leave Protocol. Step 1: Each sponsor Msi in Tsi for i ∈ [1, k] • broadcasts tree BT(si) BT(s ) k Msi −− − −i − −→ ∪i=1 Ci Step 2: Each member • updates key tree by merging all trees, • removes all keys and bkeys from the sponsor node, The sponsor Ms (additionally) • generates new share rs and computes brs, • computes all keys and bkeys from its parent to the node just below root, • broadcasts updated tree BT(s) k BT(s) M ∪i=1 Ci ←− − − − − s Step 3: Each member computes group key using BT(s) c.
Leave Protocol. The leave protocol is initiated when a valid member becomes invalid, for instance, when the group subscription time expires or if a valid member unsubscribes from the group membership. In either case, the sponsor member of the group is responsible for initiating the leave protocol. Here, 𝑁𝑗 is the sponsor of the group, and 𝑁𝑖 is the departing member. Upon the occurrence of a leave event, the sponsor 𝑁𝑖 initiates the leave protocol, which proceeds as follows:
Leave Protocol. Step 1 : Every member – update key tree by removing the leaving member node, – remove relevant parent node, if this node have only one member node, – remove all keys and bkeys from the leaf node related to the sponsor to the root node. The sponsor Ms additionally – generates new share and computes all [key, bkey] pairs on the key-path, – broadcasts updated tree T^s including only bkeys. −−−−−−−−−−−−−→ C − L T (BK ) Ms ^ s s ∗
Leave Protocol. Once again, we start with n members and assume that member Md leaves the group. The sponsor in this case is the rightmost leaf node of the subtree rooted at the leaving member’s sibling node. First off, as shown in Figure 4, each member updates its key tree by deleting the leaf node corresponding to Md. The former sibling of Md is promoted to replace Md’s parent node. The sponsor generates a new key share, computes all [key, bkey] pairs on the key path up to the root, and broadcasts the new set of bkeys. This allows all members to compute the new group key. ^ Looking at the setting in Figure 5, if member M3 leaves the group, every remaining member deletes (1, 1) and (2, 2). After updating the tree, the sponsor (M5) picks a new share K(2,3), recomputes K(1,1), K(0,0), BK(2,3) and BK(1,1), and broadcasts the updated tree T5 with BK5∗. Upon receiving the broadcast message, all members compute the group key. Note that M3 cannot compute the group key, though it knows all the bkeys, because its share is no longer part of the group key.
Leave Protocol. The leave protocol operates when any user needs to leave the group. A leaving member sends a leave request message to a director. For the proposed scheme, the director is designed to be a member below the leaving node in a tree in order to minimize the computation. In a case that the leaving node is the first member in an existing tree, the director can be the second member in the tree. The director has to compute a new key tree, and then broadcasts it to all members. The next example is continued with the scenario from the join protocol in section 3.7. In this leave protocol scenario, it is assumed that user C is going to leave the group. User B is going to be a director of the group and responsible to compute a new key tree as shown in Figure 3.4. In this case, user B computes a new public key PKAB, a new group key KABD = KAB PKD KAB -1, and a new public group key PKABD. Then user B broadcasts new key tree to all members. The proposed scheme designed to comply with the concept of forward secrecy as stated above. For example, the leaving user C cannot know the value of the new group key KABD and he cannot use his private key to decrypt messages. Next is an example that continued from the join operation as the following. The director, user B can compute key tree as the following; KABD = KAB PKD KAB -1 = (σ8σ7σ9) (σ1σ4σ3) σ9σ6σ2σ2 (σ1σ4σ3)-1 (σ8σ7σ9)-1 (σ19σ17) σ2σ2σ9σ6σ18σ16 (σ19σ17)-1(σ8σ7σ9) (σ1σ4σ3) σ2-1σ2-1σ6-1σ9-1 (σ1σ4σ3)-1 (σ8σ7σ9)-1 (8) PKABD = KABD gAgBgD KABD-1 16 18 = (σ8σ7σ9) (σ1σ4σ3) σ9σ6σ2σ2 (σ1σ4σ3)-1 (σ8σ7σ9)-1 (σ19σ17) σ2σ2σ9σ6σ18σ16 (σ19σ17)-1(σ8σ7σ9) (σ1σ4σ3) σ2-1σ2-1σ6-1σ9-1 (σ1σ4σ3)-1 (σ8σ7σ9)-1 σ2σ2σ9σ6 σ18σ16 (σ8σ7σ9) (σ1σ4σ3) σ9σ6σ2σ2 (σ1σ4σ3)-1(σ8σ7σ9)-1(σ19σ17) σ -1σ -1σ6-1σ9-1σ2-1σ2-1(σ19σ17)-1(σ8σ7σ9) (σ1σ4σ3) σ2-1σ2-1σ6-1σ9-1(σ1σ4σ3)-1 (σ8σ7σ9)-1 The value of KABD can be computed by user D as the following; KABD = d PKAB d-1 = (σ19σ17) (σ8σ7σ9) (σ1σ4σ3) σ9σ6σ2σ2 (σ1σ4σ3)-1(σ8σ7σ9)-1 σ2σ2σ9σ6σ18σ16 (σ8σ7σ9) (σ1σ4σ3) σ2-1σ2-1σ6-1σ9-1(σ1σ4σ3)-1(σ8σ7σ9)-1(σ19σ17)-1 (9) We can see that (8) = (9)
