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Here we use combined GDH and TGDH for secure group sharing.
TGDH is a distributed protocol, it requires no centralized entity to transport secret keys.
The Key Agreement Protocol (Combined GDH and TGDH) is implemented among the group members.
In the setup operation, LDGKA requires more exponentiations than TGDH; however, in the join or leave operations, LDGKA does not require any exponentiations at all, while the others still require exponentiations to complete the operations.
Communication rounds Computations (exps) Pure GDH N ([N.sup.2] + 3N)/2-1 TGDH [Log.sub.2]W 2[Log.sub.2]N-1 LDGKA [Log.sub.2]N-[Log.sub.2]m 2m[Log.sub.2]N-2m[Log.sub.2]m-m TABLE 3: Comparison of the join operation.
Table 1 shows summary of comparison with other DH-based key management protocols: CKD, GDH, BD, STR, and TGDH. In Table 1, n, m, and p denote the number of current group members, joining or merged-group members, and leaving or partitioned-group members, respectively.
STR and TGDH are tree-based key agreement protocols.
TGDH was proposed to increase the efficiency of the GK generation process by using a tree structure .
To generate a GK efficiently in TGDH, a group controller must exist to manage the overall GK generation process.
The number of transferred keys model join leave multicast unicast multicast unicast BD  2n 0 2n 0 GDH  n n - 1 n log n - 1 TGDH  log n log n + 1 log n 0 D-LKH  2log n log n + 1 2 log n - 1 0 D-OFT  log n log n + 1 log n 0 Proposed 0 log n + 2 2 0 Table 2.
In this section the proposed protocol (ETGDH) is compared to other contributory group key agreement schemes including TGDH, GDH, BD, and STR.
The tests designed were run on the testbed in order to measure the average response times of computing a group key and the communication overhead of sending and delivering messages under the selected protocols, which included GDH (Group Diffie-Hellman) , BD (Burmester and Desmedt) , STR (Skinny Tree) , and TGDH (Tree-based Group Diffie-Hellman) .
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