Abstract. We construct the first public-key encryption scheme that is proven secure (in the standard model, under standard assumptions) even when the attacker gets access to encryptions of arbitrary efficient functions of the secret key. Specifically, under either the DDH or LWE assumption, and for arbitrary but fixed polynomials L and N , we obtain a public-key encryption scheme that resists key-dependent message (KDM) attacks for up to N (k) public keys and functions of circuit size up to L(k), where k denotes the size of the secret key. We call such a scheme bounded KDM secure. Moreover, we show that our scheme suffices for one of the important applications of KDM security: ability to securely instantiate symbolic protocols with axiomatic proofs of security.We also observe that any fully homomorphic encryption scheme that additionally enjoys circular security and circuit privacy is fully KDM secure in the sense that its algorithms can be independent of the polynomials L and N as above. Thus, the recent fully homomorphic encryption scheme of Gentry (STOC 2009) is fully KDM secure under certain non-standard hardness assumptions.Finally, we extend an impossibility result of Haitner and Holenstein (TCC 2009), showing that it is impossible to prove KDM security against a family of query functions that contains exponentially hard pseudorandom functions if the proof makes only a black-box use of the query function and the adversary attacking the scheme. This shows that the non-black-box use of the query function in our proof of security is inherent.

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We study the possibility of constructing encryption schemes secure under messages that are chosen depending on the key k of the encryption scheme itself. We give the following separation results that hold both in the private and in the public key settings:-Let H be the family of poly(n)-wise independent hash-functions. There exists no fully-black-box reduction from an encryption scheme secure against key-dependent messages to one-way permutations (and also to families of trapdoor permutations) if the adversary can obtain encryptions of h(k) for h ∈ H.-There exists no reduction from an encryption scheme secure against key-dependent messages to, essentially, any cryptographic assumption, if the adversary can obtain an encryption of g(k) for an arbitrary g, as long as the reduction's proof of security treats both the adversary and the function g as black boxes.

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