Table 1: Communication and computation costs.Rounds Messages Exponentiations Signatures VerificationsGDHJoin, merge m+3 n+2m+1 n+2m+1 m+3 n+2m+1
Leave 1 1 푛−푝 11
Partition 2 p+1 푛−푝 2 p
Master leave 2 푛−1 푛−1 2 푛−1STRJoin 3 m+2 3 m 2 m+2
Leave 1 1 3(푛 − 푝) − 2푠 − 1 11
Merge 3 3 3푚 − 1 23
Partition 2 3 3(푛 − 푝) − 2푠 − 1 23TGDHJoin, merge 2 3 3ℎ − 3 23
Leave, partition h 2 h 3ℎ h ℎBDJoin, merge 2 2 n+2m 32 2푛 + 2푚 − 2
Leave 2 2푛 − 2푝 32 2푛−2푝−2
Partition 2 2 n 32 2푛−2푝−2CKDJoin, merge 3 m+2 n+2m 3 m+2
Leave 1 1 푛−푝 11
Partition 3 p+2 max(푛 − 푝,2푝−1) 34
Master leave 3 n 2푛 − 3 3 nCODHJoin 2 m+1 n+m+1 2 m+1
Leave 1 1 푛−푝 11
Merge 2 2 푛+푚 22
Partition 2 3 푛−푝 22
Master leave 2 2 푛−1 22Table 2: Communication and computation costs for CODH member and master.Send Receive Exponentiations Signatures VerificationsGeneral member Join, merge^01101
Leave, partition 0 1 1 0 1Group masterJoin 1 mn+m 11
Leave, partition 1 0 푛−푝 10
Merge 1 1 n+m 11Accordingly, the performance of STR depends on the location
of the sponsors. In TGDH, the costs depend on the height
of the resulting key tree and locations of joining or leaving
membersinthetree.Weprovidetheworstcasecostfor
TGDH.
Most of the cost in CODH comes from the master
node. A general node consumes only one communication,
modular exponentiation, signature, and verification in all
ofgrouprekeyingprocess.Wesummarizethecostsfora
general member and the group master inTable 2.Although
the exponentiation cost looks heavy in the master, its cost
is insignificant. We conducted an experiment to measure
computation delays for modular exponentiations.Table 3
shows the average delay of 10 experimental results for each.
The first device has less CPU power than the second device.
When modular prime푝is 1024 bits long and푛≤50,the
computationdelayislessthan1s.Theaveragedelayofone
exponentiation is less than 8 ms in the second device. More-
over, reducing communication cost is important for mobile
devices because data communication consumes more energy
than any other process. Therefore, our group key protocols
can be efficiently applied in dynamic mobile networks.5. Security
Let푝be a large prime number of the form2푞 + 1for a prime
푞inZ푝.Let퐺be a cyclic group of prime order푞and let푔
be a generator of퐺;thatis,퐺=⟨푔⟩. The decisional Diffie-
Hellmanproblem(DDH)isasfollows:given(푔,푔푥,푔푦,푔푧),
where푥, 푦, 푧 ∈Z푞,decidewhether푧=푥푦or a randomly
chosen number. In particular, the security of our protocol
is based on the divisible decisional Diffie-Hellman problem