However, no known work considered the functionality requirement in its most realistic setting. We propose a new security definition of BAEKS in the multi-user (with adaptive corruptions) and multi-challenge (both in terms of ciphertext and trapdoor in an interleaved manner) settings. We also study the question of the unforgeability of BAEKS. In fact, our strong hiding requirement already implies a significant amount of unforgeability. We then propose a new BAEKS construction in the bilinear pairing groups. We prove this scheme achieves adaptive tight full-hiding security under (almost) standard MDDH assumptions. Restricting our BAEKS construction for a single receiver also gives an efficient and tightly secure PAEKS construction. We further run experiments to implement and evaluate our scheme.

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