# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020098

## Efficient fully CCA-secure predicate encryptions from pair encodings

 1 Applied Statistics Unit, Indian Statistical Institute Kolkata, West Bengal-700108, India 2 Department of Computer Science and Engineering, Indian Institute of Information Technology, Sri City, Chittoor, Andhra Pradesh - 517 646, India

* Corresponding author: Tapas Pandit

Received  January 2020 Published  July 2020

Attrapadung (Eurocrypt 2014) proposed a generic framework for fully (adaptively) CPA-secure predicate encryption (PE) based on a new primitive, called pair encodings. Following the CCA conversions of Yamada et al. (PKC 2011, 2012) and Nandi et al. (ePrint Archive: 2015/457, AAECC 2018), one can have CCA-secure PE from CPA-secure PE if the primitive PE has either verifiability or delegation. These traditional approaches degrade the performance of the resultant CCA-secure PE scheme as compared to the primitive CPA-secure PE. As an alternative, we provide a direct fully secure CCA-construction of PE from the pair encoding scheme. This costs an extra computation of group element in encryption, three extra pairing computations and one re-randomization of key in decryption as compared to the CPA-construction of Attrapadung.

Recently, Blömer et al. (CT-RSA 2016) proposed a direct CCA-secure construction of predicate encryptions from pair encodings. Although they did not use the aforementioned traditional approaches, a sort of verifiability checking is still involved in the CCA-decryption. The number of pairing computations for this checking is nearly equal to the number of paring computations in CPA-decryption. Therefore, the performance of our direct CCA-secure PE is far better than Blömer et al.

Citation: Mridul Nandi, Tapas Pandit. Efficient fully CCA-secure predicate encryptions from pair encodings. Advances in Mathematics of Communications, doi: 10.3934/amc.2020098
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##### References:
Experiment for confidentiality (adaptive-predicate IND-CCA Security)
]) used in the construction of unbounded KP-ABE with large universes">Figure 2.  A brief description of the pair encoding scheme (Scheme 4 of [1]) used in the construction of unbounded KP-ABE with large universes
The description of the first $(3q_1 + 6)$-hybrid games used in the security proof, where alt-keys are answered by ${\sf AltKeyGen}( {\sf CT}_j, x_j)$. The rest of the games are described in Figure 4
The description of the last $(2\nu + 1)$-hybrid games used in the security proof, where the key queries are answered in a way similar to $\mathrm{Game}_{1 \mbox{-} (q_1+1) \mbox{-} 3}$
A comparison between the performance (viz., the number of pairing computations) of the decryption of our construction and that of the construction of Blömer et al
 Additional Decryption Cost (number of pairing) Blömer et al [6] PES of [1] PE Scheme Features Verf (V) Other (O) Total (V+O) Our PES 1 IBE ER 4 6 10 2 PES 3 KP-FE RL $3\ell+7$ $2\ell + 8$ $5\ell + 15$ 2 PES 4 KP-ABE UnLU $4|S| + 5$ $2|S| + 7$ $6|S| + 12$ 2 PES 5 KP-ABE SC $8$ $9$ $17$ 2 PES 6 KP-DSE DSE $(n+2)|S| + 6$ $2|S| + 7$ $(n+4)|S| + 13$ 2 PES 7 CP-FE RL $6m + 7$ $3m+5$ $9m + 12$ 2 PES 8 KP-ABE SU $2|S|$ $|S| + 4$ $3|S| + 4$ 2 PES 9 KP-ABE SU $2|S|$ $|S| + 4$ $3|S| + 4$ 2 PES 10 CP-ABE SU $2\ell + 1$ $2\ell + 4$ $4\ell + 5$ 2 PES 11 CP-ABE SU $2\ell + 1$ $2\ell + 5$ $4\ell + 6$ 2 PES 12 KP-ABE LU $\mathcal{O}(|B|)$ $|S| + 4$ $\mathcal{O}(|B|) + |S| + 4$ 2 PES 13 CP-ABE LU $2\ell + \mathcal{O}(|B|)$ $|\ell + 5$ $\mathcal{O}(|B|) + 3\ell + 5$ 2 PES 14 DSE DSE $n+2$ $d+2$ $n + d + 2$ 2
 Additional Decryption Cost (number of pairing) Blömer et al [6] PES of [1] PE Scheme Features Verf (V) Other (O) Total (V+O) Our PES 1 IBE ER 4 6 10 2 PES 3 KP-FE RL $3\ell+7$ $2\ell + 8$ $5\ell + 15$ 2 PES 4 KP-ABE UnLU $4|S| + 5$ $2|S| + 7$ $6|S| + 12$ 2 PES 5 KP-ABE SC $8$ $9$ $17$ 2 PES 6 KP-DSE DSE $(n+2)|S| + 6$ $2|S| + 7$ $(n+4)|S| + 13$ 2 PES 7 CP-FE RL $6m + 7$ $3m+5$ $9m + 12$ 2 PES 8 KP-ABE SU $2|S|$ $|S| + 4$ $3|S| + 4$ 2 PES 9 KP-ABE SU $2|S|$ $|S| + 4$ $3|S| + 4$ 2 PES 10 CP-ABE SU $2\ell + 1$ $2\ell + 4$ $4\ell + 5$ 2 PES 11 CP-ABE SU $2\ell + 1$ $2\ell + 5$ $4\ell + 6$ 2 PES 12 KP-ABE LU $\mathcal{O}(|B|)$ $|S| + 4$ $\mathcal{O}(|B|) + |S| + 4$ 2 PES 13 CP-ABE LU $2\ell + \mathcal{O}(|B|)$ $|\ell + 5$ $\mathcal{O}(|B|) + 3\ell + 5$ 2 PES 14 DSE DSE $n+2$ $d+2$ $n + d + 2$ 2
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