In a searchable encryption scheme based on the certificateless cryptosystem, the trapdoor generation algorithm only has a hash function that outputs an integer value. Does the scheme satisfy trapdoor indistinguishability? If so, how can we prove it?
Although without a reference, from your description the answer is likely no. For trapdoor indistinguishability (in public key searchable encryption, since you mentioned certficateless), the client must have some secret information that is used in generating the trapdoor. If the trapdoor is generated by a hash function (without a key), then the adversary can easily distinguish trapdoors by hashing a list of keywords locally and comparing them to the trapdoors.