In this subsection, we introduce two lemmas to demonstrate that our scheme is strongly unforgeable against PKR attacks and MKGC attacks based on the CDH, DL and CRHF assumptions defined in Section 2.
Our scheme is strongly secure against MKGC attacks launched by [A.sub.2] assuming BSW is strongly existential unforgeable.
Here, in order to launch MKGC attacks more easily, [A.sub.2] is allowed to set some trapdoors during the initialization phase.
In general, we demonstrate that our CLS scheme is strongly unforgeable against PKR attacks and MKGC attacks in the standard model by combining Lemmas 4 and 5.
To the best of our knowledge, we are the first to prove a CLS scheme to be strongly secure against the MKGC attacks without random oracles.