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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?

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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.

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  • $\begingroup$ What if the client has a symmetric key shared with the data owner beforehand, encrypts the keyword using the said symmetric key, and hashes the encrypted result as the trapdoor? $\endgroup$ Mar 25, 2022 at 4:29
  • $\begingroup$ Then it may be OK. Most of the time, trapdoor indistinguishability is defined as a game in which the adversary can query the trapdoor oracles with constraints that the challenge keywords cannot be queried. If the keywords are encrypted before hashed, and assuming the adversary cannot access the encryption key, it should be easy to prove this property. $\endgroup$ Mar 25, 2022 at 11:57
  • $\begingroup$ How can I prove, though? Because most of the proofs are based on games that involve breaking hard problems like DL, CDH, DDH, BDH. $\endgroup$ Mar 26, 2022 at 4:05
  • $\begingroup$ "encrypts the keyword using the said symmetric key" so likely you can reduce to the security of encryption scheme (modelled as a pseudorandom function/permutation), i.e. if it is not trapdoor indistinguishable then the encryption scheme is not secure pseudorandom function/permutation. $\endgroup$ Mar 27, 2022 at 9:32
  • $\begingroup$ Can you give an example? Is there a paper you can cite? $\endgroup$ Mar 27, 2022 at 10:48

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