One of the primary research challenges in Attribute-Based Encryption
(ABE) is constructing and proving cryptosystems that are adaptively
secure. To date the main paradigm for achieving adaptive security in
ABE is dual system encryption. However, almost all such solutions in
bilinear groups rely on (variants of) either the subgroup decision
problem over composite order groups or the decision linear assumption.
Both of these assumptions are decisional rather than search
assumptions and the target of the assumption is a source or bilinear
group element. This is in contrast to earlier selectively secure ABE
systems which can be proven secure from either the decisional or
search Bilinear Diffie-Hellman assumption. In this work we make
progress on closing this gap by giving a new ABE construction for the
subset functionality and prove security under the Search Bilinear
Diffie-Hellman assumption.

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We first provide a framework for
proving adaptive security in Attribute-Based Encryption systems. We
introduce a concept of ABE with deletable attributes where any party
can take a ciphertext encrypted under the attribute string $x in {0,
1}^n$ and modify it into a ciphertext encrypted under any string $x’
in {0, 1, bot}^n$ where $x’$ is derived by replacing any bits of
$x$ with $bot$ symbols (i.e. “deleting” attributes of $x$). The
semantics of the system are that any private key for a circuit $C$ can
be used to decrypt a ciphertext associated with $x’$ if none of the
input bits read by circuit $C$ are $bot$ symbols and $C(x’) = 1$.

We show a pathway for combining ABE
with deletable attributes with constrained psuedorandom functions to
obtain adaptively secure ABE building upon the recent work of
Tsabary. Our new ABE system will be adaptively
secure and be a ciphertext-policy ABE that supports the same
functionality as the underlying constrained PRF as long as the PRF is
“deletion conforming”. Here we also provide a simple constrained PRF
construction that gives subset functionality.

Our approach enables us to access a
broader array of Attribute-Based Encryption schemes support deletion
of attributes. For example, we show that both the Goyal~et
al.~(GPSW) and Boyen ABE schemes can
trivially handle a deletion operation. And, by using a hardcore bit
variant of GPSW scheme we obtain an adaptively secure ABE scheme under
the Search Bilinear Diffie-Hellman assumption in addition to
pseudo random functions in NC1. This gives the first adaptively
secure ABE from a search assumption as all prior work relied on
decision assumptions over source group elements.

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