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Predicate encryption is an advanced form of public-key encryption that yields high flexibility in terms of access control. In the literature, many predicate encryption schemes have been proposed such as fuzzy-IBE, KP-ABE, CP-ABE, doubly spatial encryption DSE, and ABE for arithmetic span programs. In this paper, we study relations among them and show that some of them are in fact equivalent by giving conversions among them. More specifically, our main contributions are as follows:We show that monotonic, small universe KP-ABE CP-ABE with bounds on the size of attribute sets and span programs or linear secret sharing matrix can be converted into DSE. Furthermore, we show that DSE implies non-monotonic CP-ABE and KP-ABE with the same bounds on parameters. This implies that monotonic/non-monotonic KP/CP-ABE with the bounds and DSE are all equivalent in the sense that one implies another.We also show that if we start from KP-ABE without bounds on the size of span programs but bounds on the size of attribute sets, we can obtain ABE for arithmetic span programs. The other direction is also shown: ABE for arithmetic span programs can be converted into KP-ABE. These results imply, somewhat surprisingly, KP-ABE without bounds on span program sizes is in fact equivalent to ABE for arithmetic span programs, which was thought to be more expressive or at least incomparable. By applying these conversions to existing schemes, we obtain many non-trivial consequences. We obtain the first non-monotonic, large universe CP-ABE that supports span programs with constant-size ciphertexts, the first KP-ABE with constant-size private keys, the first adaptively-secure, multi-use ABE for arithmetic span programs with constant-size ciphertexts, and more. We also obtain the first attribute-based signature scheme that supports non-monotone span programs and achieves constant-size signatures via our techniques.

Conversions Among Several Classes of Predicate Encryption and Applications to ABE with Various Compactness Tradeoffs