## Cryptology ePrint Archive: Report 2015/180

Key-Homomorphic Constrained Pseudorandom Functions

Abhishek Banerjee and Georg Fuchsbauer and Chris Peikert and Krzysztof Pietrzak and Sophie Stevens

Abstract: A pseudorandom function (PRF) is a keyed function $F \colon {\cal K}\times{\cal X}\rightarrow {\cal Y}$ where, for a random key $k\in{\cal K}$, the function $F(k,\cdot)$ is indistinguishable from a uniformly random function, given black-box access. A \emph{key-homomorphic} PRF has the additional feature that for any keys $k,k'$ and any input $x$, we have $F(k + k', x)= F(k,x) \oplus F(k',x)$ for some group operations $+, \oplus$ on $\cal{K}$ and $\cal{Y}$, respectively. A \emph{constrained} PRF for a family of sets ${\cal S} \subseteq \cal{P}({\cal X})$ has the property that, given any key $k$ and set $S \in \cal{S}$, one can efficiently compute a constrained'' key $k_S$ that enables evaluation of $F(k,x)$ on all inputs $x\in S$, while the values $F(k,x)$ for $x \notin S$ remain pseudorandom even given $k_{S}$.

In this paper we construct PRFs that are simultaneously constrained \emph{and} key homomorphic, where the homomorphic property holds even for constrained keys. We first show that the multilinear map-based bit-fixing and circuit-constrained PRFs of Boneh and Waters (Asiacrypt 2013) can be modified to also be \emph{key-homomorphic}. We then show that the LWE-based key-homomorphic PRFs of Banerjee and Peikert (Crypto 2014) are essentially already \emph{prefix-constrained} PRFs, using a (non-obvious) definition of constrained keys and associated group operation. Moreover, the constrained keys themselves are pseudorandom, and the constraining and evaluation functions can all be computed in low depth.

As an application of key-homomorphic constrained PRFs, we construct a proxy re-encryption scheme with fine-grained access control. This scheme allows storing encrypted data on an untrusted server, where each file can be encrypted relative to some attributes, so that only parties whose constrained keys match the attributes can decrypt. Moreover, the server can re-key (arbitrary subsets of) the ciphertexts without learning anything about the plaintexts, thus permitting efficient and fine-grained revocation.

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Original Publication (in the same form): IACR-TCC-2015