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doi: 10.3934/amc.2020098
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Efficient fully CCA-secure predicate encryptions from pair encodings

Mridul Nandi 1, and  Tapas Pandit 2,,

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 Early access 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.

Keywords: Pair encodings, predicate encryption, CCA-security, conversion, efficiency.
Mathematics Subject Classification: Primary: 94A60; Secondary: 68P25.
Citation: Mridul Nandi, Tapas Pandit. Efficient fully CCA-secure predicate encryptions from pair encodings. Advances in Mathematics of Communications, doi: 10.3934/amc.2020098
References:
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N. Attrapadung, B. Libert and E. Panafieu, Expressive key-policy attribute-based encryption with constant-size ciphertexts, In PKC, Lecture Notes in Comput. Sci., volume 6571, Springer, 2011, 90–108. doi: 10.1007/978-3-642-19379-8_6.  Google Scholar

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J. Blömer and G. Liske, Construction of fully cca-secure predicate encryptions from pair encoding schemes, In CT-RSA, Lecture Notes in Comput. Sci., volume 9610, Springer, 2016,431–447. doi: 10.1007/978-3-319-29485-8_25.  Google Scholar

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K. Emura, A. Miyaji, A. Nomura, K. Omote and M. Soshi, A ciphertext-policy attribute-based encryption scheme with constant ciphertext length, In ISPEC, Lecture Notes in Comput. Sci., volume 5451, Springer, 2009, 13–23. doi: 10.1504/IJACT.2010.033798.  Google Scholar

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show all references

References:
[1]

N. Attrapadung, Dual system encryption via doubly selective security: Framework, fully secure functional encryption for regular languages, and more, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 8441, Springer, 2014,557–577. doi: 10.1007/978-3-642-55220-5_31.  Google Scholar

[2]

N. Attrapadung, Fully secure and succinct attribute based encryption for circuits from multi-linear maps, Cryptology ePrint Archive, Report 2014/772, 2014, http://eprint.iacr.org/. Google Scholar

[3]

N. Attrapadung and B. Libert, Functional encryption for inner product: Achieving constant-size ciphertexts with adaptive security or support for negation, In PKC, Lecture Notes in Comput. Sci., volume 6056, Springer, 2010,384–402. doi: 10.1007/978-3-642-13013-7_23.  Google Scholar

[4]

N. Attrapadung, B. Libert and E. Panafieu, Expressive key-policy attribute-based encryption with constant-size ciphertexts, In PKC, Lecture Notes in Comput. Sci., volume 6571, Springer, 2011, 90–108. doi: 10.1007/978-3-642-19379-8_6.  Google Scholar

[5]

N. Attrapadung and S. Yamada, Duality in ABE: Converting attribute based encryption for dual predicate and dual policy via computational encodings, In CT-RSA, Lecture Notes in Comput. Sci., volume 9048, Springer, 2015, 87–105. doi: 10.1007/978-3-319-16715-2_5.  Google Scholar

[6]

J. Blömer and G. Liske, Construction of fully cca-secure predicate encryptions from pair encoding schemes, In CT-RSA, Lecture Notes in Comput. Sci., volume 9610, Springer, 2016,431–447. doi: 10.1007/978-3-319-29485-8_25.  Google Scholar

[7]

D. Boneh, R. Canetti, S. Halevi and J. Katz, Chosen-ciphertext security from identity-based encryption, Journal of SIAM, 36 (2007), 1301-1328.  doi: 10.1137/S009753970544713X.  Google Scholar

[8]

D. Boneh and M. Franklin, Identity-based encryption from the weil pairing, In CRYPTO, Lecture Notes in Comput. Sci., volume 2139, Springer, 2001,213–229. doi: 10.1007/3-540-44647-8_13.  Google Scholar

[9]

D. Boneh, E. Goh, and K. Nissim, Evaluating 2-dnf formulas on ciphertexts, In TCC, Lecture Notes in Comput. Sci., volume 3378, Springer, 2005,325–341. doi: 10.1007/978-3-540-30576-7_18.  Google Scholar

[10]

D. Boneh and M. Hamburg, Generalized identity-based and broadcast encryption schemes, In ASIACRYPT, Lecture Notes in Comput. Sci., volume 5350, Springer, 2008,455–470. doi: 10.1007/978-3-540-89255-7_28.  Google Scholar

[11]

D. Boneh and J. Katz, Improved efficiency for CCA-secure cryptosystems built using identity-based encryption, In CT-RSA, Lecture Notes in Comput. Sci., volume 3376, Springer, 2005, 87–103. doi: 10.1007/978-3-540-30574-3_8.  Google Scholar

[12]

X. Boyen, Q. Mei and B. Waters, Direct chosen ciphertext security from identity-based techniques, In ACM Conference on Computer and Communications Security, ACM, New York, 2005,320–329. doi: 10.1145/1102120.1102162.  Google Scholar

[13]

R. Canetti, S. Halevi and J. Katz, Chosen-ciphertext security from identity-based encryption, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 3027, Springer, 2004,207–222. doi: 10.1007/978-3-540-24676-3_13.  Google Scholar

[14]

M. Chase, Multi-authority attribute based encryption, In TCC, Lecture Notes in Comput. Sci., volume 4392, Springer, 2007,515–534. doi: 10.1007/978-3-540-70936-7_28.  Google Scholar

[15]

C. Chen, Z. Zhang and D. Feng, Efficient ciphertext policy attribute-based encryption with constant-size ciphertext and constant computation-cost, In PROVSEC, Lecture Notes in Comput. Sci., volume 6980, Springer, 2011, 84–101. doi: 10.1007/978-3-642-24316-5_8.  Google Scholar

[16]

C. Chen, Z. Zhang and D. Feng, Fully secure doubly-spatial encryption under simple assumptions, In PROVSEC, Lecture Notes in Comput. Sci., volume 7496, Springer, 2012,253–263. doi: 10.1007/978-3-642-33272-2_16.  Google Scholar

[17]

J. Chen and H. Wee, Doubly spatial encryption from DBDH, Theoret. Comput. Sci., 543 (2014), 79-89.  doi: 10.1016/j.tcs.2014.06.003.  Google Scholar

[18]

C. Cocks, An identity based encryption scheme based on quadratic residues, In Cryptography and Coding, Lecture Notes in Comput. Sci., volume 2260, Springer, 2001,360–363. doi: 10.1007/3-540-45325-3_32.  Google Scholar

[19]

W. Diffie and M. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, 22 (1976), 644-654.  doi: 10.1109/tit.1976.1055638.  Google Scholar

[20]

K. Emura, A. Miyaji, A. Nomura, K. Omote and M. Soshi, A ciphertext-policy attribute-based encryption scheme with constant ciphertext length, In ISPEC, Lecture Notes in Comput. Sci., volume 5451, Springer, 2009, 13–23. doi: 10.1504/IJACT.2010.033798.  Google Scholar

[21]

E. Fujisaki and T. Okamoto, Secure integration of asymmetric and symmetric encryption schemes, In CRYPTO, Lecture Notes in Comput. Sci., volume 1666, Springer, 1999,537–554. doi: 10.1007/3-540-48405-1_34.  Google Scholar

[22]

S. Garg, C. Gentry, S. Halevi, A. Sahai and B. Waters, Attribute-based encryption for circuits from multilinear maps, In CRYPTO, Lecture Notes in Comput. Sci., volume 8043, Springer, 2013,479–499. doi: 10.1007/978-3-642-40084-1_27.  Google Scholar

[23]

C. Gentry and A. Silverberg, Hierarchical ID-based cryptography, In ASIACRYPT, Lecture Notes in Comput. Sci., volume 2501, Springer, 2002,548–566. doi: 10.1007/3-540-36178-2_34.  Google Scholar

[24]

S. Gorbunov, V. Vaikuntanathan and H. Wee., Attribute-based encryption for circuits, In STOC'13–Proceedings of the 2013 ACM Symposium on Theory of Computing, ACM, 2013,545–554. doi: 10.1145/2488608.2488677.  Google Scholar

[25]

V. Goyal, A. Jain, O. Pandey and A. Sahai, Bounded ciphertext policy attribute based encryption, In Automata, Languages and Programming. Part II, Lecture Notes in Comput. Sci., volume 5126, Springer, 2008,579–591. doi: 10.1007/978-3-540-70583-3_47.  Google Scholar

[26]

V. Goyal, O. Pandey, A. Sahai and B. Waters, Attribute-based encryption for fine-grained access control of encrypted data, In ACM Conference on Computer and Communications Security, ACM, 2006, 89–98. doi: 10.1145/1180405.1180418.  Google Scholar

[27]

M. Hamburg, Spatial encryption, Cryptology ePrint Archive, Report 2011/389, 2011, http://eprint.iacr.org/. Google Scholar

[28]

J. Katz, A. Sahai and B. Waters, Predicate encryption supporting disjunctions, polynomial equations, and inner products, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 4965, Springer, 2008,146–162. doi: 10.1007/978-3-540-78967-3_9.  Google Scholar

[29]

A. Lewko and B. Waters, New techniques for dual system encryption and fully secure HIBE with short ciphertexts, In TCC, of Lecture Notes in Comput. Sci., volume 5978, Springer, 2010,455–479. doi: 10.1007/978-3-642-11799-2_27.  Google Scholar

[30]

A. Lewko and B. Waters, Decentralizing attribute-based encryption, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 6632, Springer, 2011,568–588. doi: 10.1007/978-3-642-20465-4_31.  Google Scholar

[31]

A. Lewko, T. Okamoto, A. Sahai, K. Takashima and B. Waters, Fully secure functional encryption: Attribute-based encryption and (hierarchical) inner product encryption, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 6110, Springer, 2010, 62–91. doi: 10.1007/978-3-642-13190-5_4.  Google Scholar

[32]

D. Moriyama and H. Doi, A fully secure spatial encryption scheme, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 94 (2011), 28-35.  doi: 10.1587/transfun.E94.A.28.  Google Scholar

[33]

M. Nandi and T. Pandit, Generic conversions from CPA to CCA secure functional encryption, Cryptology ePrint Archive, Report 2015/457, 2015, http://eprint.iacr.org/. Google Scholar

[34]

M. Nandi and T. Pandit, Verifiability-based conversion from CPA to CCA-secure predicate encryption, Appl. Algebra Engrg. Comm. Comput., 29 (2018), 77-102.  doi: 10.1007/s00200-017-0330-2.  Google Scholar

[35]

M. Nandi and T. Pandit, Delegation-based conversion from CPA to CCA-secure predicate encryption, International Journal of Applied Cryptography, 4 (2020), 16-35.  doi: 10.1504/ijact.2020.107163.  Google Scholar

[36]

T. Okamoto and K. Takashima, Hierarchical predicate encryption for inner-products, In ASIACRYPT, Lecture Notes in Comput. Sci., volume 5912, Springer, 2009,214–231. doi: 10.1007/978-3-642-10366-7_13.  Google Scholar

[37]

T. Okamoto and K. Takashima, Fully secure functional encryption with general relations from the decisional linear assumption, In CRYPTO, Lecture Notes in Comput. Sci., volume 6223, Springer, 2010,191–208. doi: 10.1007/978-3-642-14623-7_11.  Google Scholar

[38]

T. Okamoto and K. Takashima, Achieving short ciphertexts or short secret-keys for adaptively secure general inner-product encryption, In Cryptology and Network Security, Lecture Notes in Comput. Sci., volume 7092, Springer, 2011,138–159. doi: 10.1007/978-3-642-25513-7_11.  Google Scholar

[39]

T. Okamoto and K. Takashima, Adaptively attribute-hiding (hierarchical) inner product encryption, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 7237, Springer, 2012,591–608. doi: 10.1007/978-3-642-29011-4_35.  Google Scholar

[40]

T. Okamoto and K. Takashima, Fully secure unbounded inner-product and attribute-based encryption, In ASIACRYPT, Lecture Notes in Comput. Sci., volume 7658, Springer, 2012,349–366. doi: 10.1007/978-3-642-34961-4_22.  Google Scholar

[41]

R. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM, 21 (1978), 120-126.  doi: 10.1145/359340.359342.  Google Scholar

[42]

A. Sahai and B. Waters, Fuzzy identity-based encryption, In EUROCRYPT, Lecture Notes in Comput. Sci., volume 3494, Springer, 2005,457–473. doi: 10.1007/11426639_27.  Google Scholar

[43]

A. Shamir, Identity-based cryptosystems and signature schemes, In CRYPTO, Lecture Notes in Comput. Sci., volume 196, Springer, 1984, 47–53. doi: 10.1007/3-540-39568-7_5.  Google Scholar

[44]

B. Waters, Ciphertext-policy attribute-based encryption: An expressive, efficient, and provably secure realization, In PKC, Lecture Notes in Comput. Sci., volume 6571, Springer, 2011, 53–70. doi: 10.1007/978-3-642-19379-8_4.  Google Scholar

[45]

B. Waters, Dual system encryption: Realizing fully secure IBE and HIBE under simple assumptions, In CRYPTO, Lecture Notes in Comput. Sci., volume 5677, Springer, 2009,619–636. doi: 10.1007/978-3-642-03356-8_36.  Google Scholar

[46]

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Figure 1.  Experiment for confidentiality (adaptive-predicate IND-CCA Security)
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1]) 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
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Figure 3.  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
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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} $
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Table 1.  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
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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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