Case 2. In Protocol 1, case 4 was satisfied for Ppred at subround t + j′. This means Ppred received (sid, v, vk, σ) and a valid signature σ, and sent it to all of its neighbors, and in particular Pi′ would receive this message by subround t + j′ + 1. Assume Pi′ has not already output (vk, sid, v′, GradeKeyi′ (vk, sid)) for v′ ∈ {v, ⊥} (otherwise we are already done). In that case, for Pi′ it must hold that Bvk,sid = 0. By Inequality 3.1, GradeKeyi′ (vk, sid) > 0; hence, one of the following must be the case:
Case 2. 1. In the case when the in-degree of one of nodes N1, N2, N3 is two, e.g., N3, mass join is connecting J to N3. Similar to Case 1, a subgroup key KG3 ∪J for G3 ∪ J can be achieved. Next the coordinator of G3 broadcast KG1 G, KG2 G and G ∪ KG3 ∪J in G∗ = G J. Then each member in G∗ is able to derive a new group key KG∗ = k(3) 1 \ \ K K K h(eˆ(G, G) G1 G2 G3 ∪J ) / \`\
Case 2. In the case when the in-degree of the node is three, the subgroup under this node is composed of three subgroups G1, G2, G3 under three nodes, respectively. Assume that the subgroup key KGi (i = 1, 2, 3) of Gi has been established, then the coordi- nator of Gi (i = 1, 2, 3) broadcasts KGi in other subgroups if it is unknown to them. After that, each member in G1 ∪G2 ∪U3 is able to derive a new sub- group key KG1 ∪G2 ∪G3 = h(eˆ( , )KG1 KG2 KG3 ). For example, in Fig. 6, the secret key k(2) = G G is a special case of mass leave. After mass leaving, a new group key for the rest of the peer group is reached in two steps.
Case 2. In the case when the in-degree of the root node is three, the group is composed of three subgroups G1, G2, G3 under three nodes 1, 2, 3. Suppose that the subgroup key for Gi (i=1,2,3) is KG , then K(3, KG , KG , KG ) = based on DLP over Zp with p a 1024-bit prime [22][23]. In the case when n = 3, view(3, r1, r2, r3) = {r1G, r2G, r3G} and K(3, r1, r2, r3) = h(eˆ(G, G)r1 r2 r3 ).
Case 2. In the case when the in-degree of the node is three, the subgroup under this node is composed of three subgroups G1, G2, G3 under three nodes, respectively. Assume that the subgroup key KGi (i = 1, 2, 3) of Gi has been established, then the coordi- nator of Gi (i = 1, 2, 3) broadcasts KGi in other subgroups if it is unknown to them. After that, each member in G1 ∪ G2 ∪ U3 is able to derive a new sub- group key KG1 ∪G2 ∪G3 = h(eˆ( , )KG1 KG2 KG3 ). For example, in Fig. 6, the secret key k(2) = G G is a special case of mass leave. After mass leaving, a new group key for the rest of the peer group is reached in two steps. h(eˆ( , )r1 k2 k3 ) for the subgroup under 3-in- degree node k(2) can be established after U1 and U4 broadcast r1G and k(1)G, respectively, because k(1)G Step 1 . (Graph Reconstruct) After mass leaving, the key agreement graph for the rest of the group is recon- structed according to the three rules: (1) After a member leaves, remove the member node and the edge with the member node as an initial node as well; (2) If there is no member under a node, remove the node and all edges with the node as the initial node or the terminal node; (3) If the in-degree of a node is one, remove the node. After that, the key agreement graph maintained in the public server is updated accordingly. For example, in Fig. 2, after U2, U3, U5, U10, U11, U12, U13, U15 leave the peer group, the key agreement graph for the rest of the group is reconstructed as shown in Fig. 6.
Case 2. If m2 ≥ |A| we use the fact that M is weakly k-right concellative with respect to the set of generators A. In order to verify condition (P), it suffices to show conditions (i) and (ii) hold with {R(x, x') | x, x' ∈ X} replaced by the broader family [ ] , the set of all k-tuples of A. This is because for each x, x' and a / R(x, x') we may verify (i) and (ii) for R ∈ A with a ∈/ R and R(x, x') ⊆ R. Thus, we need 1 − (k + 1)e |A| k
Case 2. Xxxxxx Schemes Xxxxxx’x rule is an algorithm for polynomial computation that reduces the number of multiplications and results in a computationally efficient form [OS12]. For a polyno- mial in one variable
