Attribute-based encryption (ABE) is a cryptographic primitive which supports fine-grained access control on encrypted data, making it an appealing building block for many applications. Pair encodings (Attrapadung, EUROCRYPT 2014) are simple primitives that can be used for constructing fully secure ABE schemes associated to a predicate relative to the encoding.
We propose a generic transformation that takes any pair encoding scheme (PES) for a predicate $P$ and produces a PES for its negated predicate $bar{P}$ . This construction finally solves a problem that was open since 2015. Our techniques bring new insight to the expressivity and generality of PES and can be of independent interest. We also provide, to the best of our knowledge, the first pair encoding scheme for negated doubly spatial encryption (obtained with our transformation) and explore several other consequences of our results.

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