Collusion resistance in proxy re-encryption scheme depending on bilinear map rules
A proxy re-encryption scheme is collusion resistant, if the proxy and a delegatee are not able to recover the secret key of the delegating party.


Are you sure that definition makes sense?  After all, if proxy has the Alice->Bob reencryption key, and Bob has his private key, then the two of them can jointly decrypt any message encrypted by Alice's public key.  And, the point of a private key is to be able to decrypt messages, so (no matter what we do) the proxy and Bob effectively have the private key (even if they might come up with the same bit pattern).

So, what does collusion resistance mean in this context?  If the requirement is that they cannot determine the bit pattern of Alice's key, well, we can easily do that by using a scheme with multiple equivalent keys (and so even if they can come up with 'a' key, it is unlikely to be Alice's bit pattern).  However, that doesn't sound very useful...