February  2017, 11(1): 161-177. doi: 10.3934/amc.2017010

Forward-secure identity-based encryption with direct chosen-ciphertext security in the standard model

College of Computer and Information, Hohai University, No.8, Focheng Xi Road, Jiangning District, Nanjing, Jiangsu 211100, China

Received  August 2015 Revised  December 2015 Published  February 2017

Fund Project: This work is supported by the Nature Science Foundation of China under Grant Nos. 61272542 and 61672207, the Natural Science Foundation of Jiangsu Province Grant No. BK20161511, the Fundamental Research Funds for the Central Universities Grant No. 2016B10114, a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Technology.

The paradigm of forward security provides a promising approach to deal with the key exposure problem as it can effectively minimize the damage caused by the key exposure. In this paper, we develop a new forward-secure identity-based encryption scheme without random oracles. We formally prove that the proposed scheme is secure against adaptive chosen-ciphertext attacks in the standard model. In the proposed scheme, the running time of the private key extraction and decryption algorithms and the sizes of the user's initial private key and the ciphertext are independent on the total number of time periods, and any other performance parameter has at most log-squared complexity in terms of the total number of time periods. Compared with the previous forward-secure identity-based encryption schemes, the proposed scheme enjoys obvious advantage in the overall performance. To the best of our knowledge, it is the first forward-secure identity-based encryption scheme that achieves direct chosen-ciphertext security in the standard model.

Citation: Yang Lu, Jiguo Li. Forward-secure identity-based encryption with direct chosen-ciphertext security in the standard model. Advances in Mathematics of Communications, 2017, 11 (1) : 161-177. doi: 10.3934/amc.2017010
References:
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M. Abdalla, S. K. Miner and C. Namprempre, Forward-secure threshold signature schemes, in Proc. CT-RSA 2001, Springer-Verlag, 2001,441-456. doi: 10.1007/3-540-45353-9_32.  Google Scholar

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M. Abdalla and L. Reyzin, A new forward-secure digital signature scheme, in Proc. Asiacrypt. 2000, Springer-Verlag, 2000,116-129. doi: 10.1007/3-540-44448-3_10.  Google Scholar

[3]

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M. Bellare and S. K. Miner, A forward-secure digital signature scheme, in Proc. Crypt. 1999, Springer-Verlag, 1999,431-448. doi: 10.1007/3-540-48405-1_28.  Google Scholar

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M. Bellare and B. Yee, Forward security in private-key cryptography, in Proc. CT-RSA 2003, Springer-Verlag, 2003, 1-18. doi: 10.1007/3-540-36563-X_1.  Google Scholar

[8]

D. Boneh and X. Boyen, Efficient selective-id identity based encryption without random oracles, in Proc. Eurocrypt. 2004, Springer-Verlag, 2004,223-238. doi: 10.1007/978-3-540-24676-3_14.  Google Scholar

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D. Boneh, X. Boyen and E. J. Goh, Hierarchical identity based encryption with constant size ciphertext, in Proc. Eurocrypt. 2005, Springer-Verlag, 2005,440-456. doi: 10.1007/11426639_26.  Google Scholar

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D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing, in Proc. Crypt. 2001, Springer-Verlag, 2001,213-229. doi: 10.1007/3-540-44647-8_13.  Google Scholar

[11]

X. Boyen, H. Shacham, E. Shen and B. Waters, Forward-secure signatures with untrusted update, in Proc. ACM CCS 2006, ACM, 2006,191-200. doi: 10.1145/1180405.1180430.  Google Scholar

[12]

R. CanettiO. Goldreich and S. Halevi, The random oracle methodology, revisited, ACM J., 51 (2004), 209-218.  doi: 10.1145/1008731.1008734.  Google Scholar

[13]

R. Canetti, S. Halevi and J. Katz, A forward-secure public-key encryption scheme, in Proc. Eurocrypt. 2003, Springer-Verlag, 2003,255-271. doi: 10.1007/3-540-39200-9_16.  Google Scholar

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R. CanettiS. Halevi and J. Katz, A forward-secure public-key encryption scheme, Cryptology J., 30 (2007), 265-294.  doi: 10.1007/s00145-006-0442-5.  Google Scholar

[15]

L. Chen and Z. Cheng, Security proof of Sakai-Kasahar's identity-based encryption scheme, in Proc. Crypt. Coding 2005, Springer-Verlag, 2005,442-459. doi: 10.1007/11586821_29.  Google Scholar

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C. Gentry and A. Silverberg, Hierarchical ID-based cryptography, in Proc. Asiacrypt. 2002, Springer-Verlag, 2002,548-566. doi: 10.1007/3-540-36178-2_34.  Google Scholar

[22]

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G. Hanaoka, Y. Hanaoka and H. Imai, Parallel key-insulated public key encryption, in Proc. PKC 2006, Springer-Verlag, 2006,105-122. doi: 10.1007/11745853_8.  Google Scholar

[24]

J. Horwitz and B. Lynn, Toward hierarchical identity-based encryption, in Proc. Eurocrypt. 2002, Springer-Verlag, 2002,466-481. doi: 10.1007/3-540-46035-7_31.  Google Scholar

[25]

G. Itkis and L. Reyzin, Forward-secure signatures with optimal signing and verifying, in Proc. Crypt. 2001, Springer-Verlag, 2001,499-514. doi: 10.1007/3-540-44647-8_20.  Google Scholar

[26]

G. Itkis and L. Reyzin, SiBIR: Signer-base intrusion-resilient signatures, in Proc. Crypt. 2002, Springer-Verlag, 2002,499-514. doi: 10.1007/3-540-45708-9_32.  Google Scholar

[27]

E. Kiltz and Y. Vahlis, CCA2 secure IBE: standard model efficiency through authenticated symmetric encryption, in Proc. CT-RSA 2008, Springer-Verlag, 2008,221-238. doi: 10.1007/978-3-540-79263-5_14.  Google Scholar

[28]

A. Kozlov and L. Reyzin, Forward-secure signatures with fast key update, in Proc. SCN 2002, Springer-Verlag, 2002,247-262. doi: 10.1007/3-540-36413-7_18.  Google Scholar

[29]

H. Krawczyk, Simple forward-secure signatures from any signature scheme, in Proc. ACM CCS 2000, ACM, 2000,108-115. doi: 10.1145/352600.352617.  Google Scholar

[30]

J. Li, F. Zhang and Y. Wang, A strong identity-based key-insulated cryptosystem, in Proc. EUC Workshops 2006, Springer-Verlag, 2006,352-361. doi: 10.1007/11807964_36.  Google Scholar

[31]

B. Libert, J. Quisquater and M. Yung, Forward-secure signatures in untrusted update environments, in Proc. ACM CCS 2007, ACM, 2007,266-275. doi: 10.1145/1315245.1315279.  Google Scholar

[32]

Y. Lu and J. G. Li, A practical forward-secure public-key encryption scheme, Networks J., 6 (2011), 1254-1261.  doi: 10.4304/jnw.6.9.1254-1261.  Google Scholar

[33]

Y. Lu and J. G. Li, Generic construction of forward-secure identity-based encryption, Computers J., 7 (2012), 3068-3074.  doi: 10.4304/jcp.7.12.3068-3074.  Google Scholar

[34]

Y. Lu and J. G. Li, New forward-secure public-key encryption without random oracles, Int. J. Comp. Math. , 90 (2013), 2603-2613. doi: 10.1080/00207160.2013.807915.  Google Scholar

[35]

Y. Lu and J. G. Li, An improved certificateless strong key-insulated signature scheme in the standard model, Adv. Math. Commun., 9 (2015), 353-373.  doi: 10.3934/amc.2015.9.353.  Google Scholar

[36]

T. Malkin, D. Micciancio and S. K. Miner, Efficient generic forward-secure signatures with an unbounded number of time periods, in Proc. Eurocrypt. 2002, Springer-Verlag, 2002,400-417. doi: 10.1007/3-540-46035-7_27.  Google Scholar

[37]

A. Shamir, Identity-based cryptosystems and signature schemes, in Proc. Crypt. 1984, Springer-Verlag, 1984, 47-53. doi: 10.1007/3-540-39568-7_5.  Google Scholar

[38]

K. Singh and N. Trichy, Lattice forward-secure identity based encryption scheme, J. Internet Serv. Inf. Sec., 2 (2012), 118-128.   Google Scholar

[39]

Z. WanX. LaiJ. WengS. LiuY. Long and X. Hong, Certificateless key-insulated signature without random oracles, J. Zhejiang Univ. Sci. A, 10 (2009), 1790-1800.  doi: 10.1631/jzus.A0820714.  Google Scholar

[40]

Z. WanX. Meng and X. Hong, Certificateless strong key-insulated signature without random oracles, J. Shanghai Jiaotong Univ. (Sci), 16 (2011), 571-576.  doi: 10.1007/s12204-011-1191-7.  Google Scholar

[41]

B. Waters, Efficient identity-based encryption without random oracles, in Proc. Eurocrypt. 2005, Springer-Verlag, 2005,114-127. doi: 10.1007/11426639_7.  Google Scholar

[42]

J. Weng, X. Li, K. F. Chen and S. L. Liu, Identity-based parallel key-insulated encryption without random oracles, in Proc. Indocrypt. 2006, Springer-Verlag, 2006,409-423. doi: 10.1007/11941378_29.  Google Scholar

[43]

J. Weng, S. L. Liu, K. F. Chen, D. Zheng and W. D. Qiu, Identity-based threshold keyinsulated encryption without random oracles, in Proc. CT-RSA 2008, Springer-Verlag, 2008,203-220. doi: 10.1007/978-3-540-79263-5_13.  Google Scholar

[44]

H. YangS. Sun and H. Li, Forward-secure identity-based encryption scheme (in Chinese), J. Univ. Electr. Sci. Techn. China, 36 (2007), 534-537.   Google Scholar

[45]

D. Yao, N. Fazio, Y. Dodis and A. Lysyanskaya, ID-based encryption for complex hierarchies with applications to forward security and broadcast encryption, in Proc. ACM CCS 2004, ACM, 2004,354-363. doi: 10.1145/1030083.1030130.  Google Scholar

[46]

J. Yu, R. Hao, H. Zhao, M. Shu and J. Fan, IRIBE: Intrusion-resilient identity-based encryption, Inf. Sci. , 329 (2016), 90-104. doi: 10.1016/j.ins.2015.09.020.  Google Scholar

[47]

J. YuF. Y. KongX. G. ChengR. Hao and J. X. Fan, Forward-secure identity-based publickey encryption without random oracles, Fundam. Inf., 111 (2011), 241-256.   Google Scholar

[48]

J. YuF. Y. KongX. G. ChengR. Hao and J. X. Fan, intrusion-resilient identity-based signature: security definition and construction, J. Syst. Softw., 85 (2012), 382-391.  doi: 10.1016/j.jss.2011.08.034.  Google Scholar

[49]

J. Yu, F. Y. Kong, X. G. Cheng, R. Hao and G. W. Li, Construction of yet another forwardsecure signature scheme using bilinear maps, in Proc. ProvSec 2008, Springer-Verlag, 2008, 83-97. doi: 10.1007/978-3-540-88733-1_6.  Google Scholar

show all references

References:
[1]

M. Abdalla, S. K. Miner and C. Namprempre, Forward-secure threshold signature schemes, in Proc. CT-RSA 2001, Springer-Verlag, 2001,441-456. doi: 10.1007/3-540-45353-9_32.  Google Scholar

[2]

M. Abdalla and L. Reyzin, A new forward-secure digital signature scheme, in Proc. Asiacrypt. 2000, Springer-Verlag, 2000,116-129. doi: 10.1007/3-540-44448-3_10.  Google Scholar

[3]

R. Anderson, Two Remarks on public key cryptology, in 4th ACM Conf. Comp. Commun. Secur. , 1997. Google Scholar

[4]

M. Bellare and S. K. Miner, A forward-secure digital signature scheme, in Proc. Crypt. 1999, Springer-Verlag, 1999,431-448. doi: 10.1007/3-540-48405-1_28.  Google Scholar

[5]

M. Bellare and A. Palacio, Protecting against key-exposure: strongly key-insulated encryption with optimal threshold, Appl. Algebra Engin. Commun. Comp., 16 (2006), 379-396.  doi: 10.1007/s00200-005-0183-y.  Google Scholar

[6]

M. Bellare and P. Rogaway, Random oracles are practical: a paradigm for designing efficient protocols, in Proc. ACM CCS 1993, ACM, 1993, 62-73. doi: 10.1145/168588.168596.  Google Scholar

[7]

M. Bellare and B. Yee, Forward security in private-key cryptography, in Proc. CT-RSA 2003, Springer-Verlag, 2003, 1-18. doi: 10.1007/3-540-36563-X_1.  Google Scholar

[8]

D. Boneh and X. Boyen, Efficient selective-id identity based encryption without random oracles, in Proc. Eurocrypt. 2004, Springer-Verlag, 2004,223-238. doi: 10.1007/978-3-540-24676-3_14.  Google Scholar

[9]

D. Boneh, X. Boyen and E. J. Goh, Hierarchical identity based encryption with constant size ciphertext, in Proc. Eurocrypt. 2005, Springer-Verlag, 2005,440-456. doi: 10.1007/11426639_26.  Google Scholar

[10]

D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing, in Proc. Crypt. 2001, Springer-Verlag, 2001,213-229. doi: 10.1007/3-540-44647-8_13.  Google Scholar

[11]

X. Boyen, H. Shacham, E. Shen and B. Waters, Forward-secure signatures with untrusted update, in Proc. ACM CCS 2006, ACM, 2006,191-200. doi: 10.1145/1180405.1180430.  Google Scholar

[12]

R. CanettiO. Goldreich and S. Halevi, The random oracle methodology, revisited, ACM J., 51 (2004), 209-218.  doi: 10.1145/1008731.1008734.  Google Scholar

[13]

R. Canetti, S. Halevi and J. Katz, A forward-secure public-key encryption scheme, in Proc. Eurocrypt. 2003, Springer-Verlag, 2003,255-271. doi: 10.1007/3-540-39200-9_16.  Google Scholar

[14]

R. CanettiS. Halevi and J. Katz, A forward-secure public-key encryption scheme, Cryptology J., 30 (2007), 265-294.  doi: 10.1007/s00145-006-0442-5.  Google Scholar

[15]

L. Chen and Z. Cheng, Security proof of Sakai-Kasahar's identity-based encryption scheme, in Proc. Crypt. Coding 2005, Springer-Verlag, 2005,442-459. doi: 10.1007/11586821_29.  Google Scholar

[16]

C. Cocks, An identity based encryption scheme based on quadratic residues, in Proc. Crypt. Coding 2001, Springer-Verlag, 2001,360-363. doi: 10.1007/3-540-45325-3_32.  Google Scholar

[17]

W. DiffieP. C. Van-Oorschot and M. J. Weiner, Authentication and authenticated key exchanges, Des. Codes Crypt., 2 (1992), 107-125.  doi: 10.1007/BF00124891.  Google Scholar

[18]

Y. Dodis, M. Franklin, J. Katz, A. Miyaji and M. Yung, Intrusion-resilient public-key encryption, in Proc. CT-RSA 2003, Springer-Verlag, 2003, 19-32. doi: 10.1007/3-540-36563-X_2.  Google Scholar

[19]

Y. Dodis, J. Katz, S. Xu and M. Yung, Key-insulated public-key cryptosystems, in Proc. Eurocrypt. 2002, Springer-Verlag, 2002, 65-82. doi: 10.1007/3-540-46035-7_5.  Google Scholar

[20]

C. Gentry, Practical identity-based encryption without random oracles, in Proc. Eurocrypt. 2006, Springer-Verlag, 2006,445-464. doi: 10.1007/11761679_27.  Google Scholar

[21]

C. Gentry and A. Silverberg, Hierarchical ID-based cryptography, in Proc. Asiacrypt. 2002, Springer-Verlag, 2002,548-566. doi: 10.1007/3-540-36178-2_34.  Google Scholar

[22]

C. G. Günther, An identity-based key-exchange protocol, in Proc. Eurocrypt. 1989, SpringerVerlag, 1990, 29-37. Google Scholar

[23]

G. Hanaoka, Y. Hanaoka and H. Imai, Parallel key-insulated public key encryption, in Proc. PKC 2006, Springer-Verlag, 2006,105-122. doi: 10.1007/11745853_8.  Google Scholar

[24]

J. Horwitz and B. Lynn, Toward hierarchical identity-based encryption, in Proc. Eurocrypt. 2002, Springer-Verlag, 2002,466-481. doi: 10.1007/3-540-46035-7_31.  Google Scholar

[25]

G. Itkis and L. Reyzin, Forward-secure signatures with optimal signing and verifying, in Proc. Crypt. 2001, Springer-Verlag, 2001,499-514. doi: 10.1007/3-540-44647-8_20.  Google Scholar

[26]

G. Itkis and L. Reyzin, SiBIR: Signer-base intrusion-resilient signatures, in Proc. Crypt. 2002, Springer-Verlag, 2002,499-514. doi: 10.1007/3-540-45708-9_32.  Google Scholar

[27]

E. Kiltz and Y. Vahlis, CCA2 secure IBE: standard model efficiency through authenticated symmetric encryption, in Proc. CT-RSA 2008, Springer-Verlag, 2008,221-238. doi: 10.1007/978-3-540-79263-5_14.  Google Scholar

[28]

A. Kozlov and L. Reyzin, Forward-secure signatures with fast key update, in Proc. SCN 2002, Springer-Verlag, 2002,247-262. doi: 10.1007/3-540-36413-7_18.  Google Scholar

[29]

H. Krawczyk, Simple forward-secure signatures from any signature scheme, in Proc. ACM CCS 2000, ACM, 2000,108-115. doi: 10.1145/352600.352617.  Google Scholar

[30]

J. Li, F. Zhang and Y. Wang, A strong identity-based key-insulated cryptosystem, in Proc. EUC Workshops 2006, Springer-Verlag, 2006,352-361. doi: 10.1007/11807964_36.  Google Scholar

[31]

B. Libert, J. Quisquater and M. Yung, Forward-secure signatures in untrusted update environments, in Proc. ACM CCS 2007, ACM, 2007,266-275. doi: 10.1145/1315245.1315279.  Google Scholar

[32]

Y. Lu and J. G. Li, A practical forward-secure public-key encryption scheme, Networks J., 6 (2011), 1254-1261.  doi: 10.4304/jnw.6.9.1254-1261.  Google Scholar

[33]

Y. Lu and J. G. Li, Generic construction of forward-secure identity-based encryption, Computers J., 7 (2012), 3068-3074.  doi: 10.4304/jcp.7.12.3068-3074.  Google Scholar

[34]

Y. Lu and J. G. Li, New forward-secure public-key encryption without random oracles, Int. J. Comp. Math. , 90 (2013), 2603-2613. doi: 10.1080/00207160.2013.807915.  Google Scholar

[35]

Y. Lu and J. G. Li, An improved certificateless strong key-insulated signature scheme in the standard model, Adv. Math. Commun., 9 (2015), 353-373.  doi: 10.3934/amc.2015.9.353.  Google Scholar

[36]

T. Malkin, D. Micciancio and S. K. Miner, Efficient generic forward-secure signatures with an unbounded number of time periods, in Proc. Eurocrypt. 2002, Springer-Verlag, 2002,400-417. doi: 10.1007/3-540-46035-7_27.  Google Scholar

[37]

A. Shamir, Identity-based cryptosystems and signature schemes, in Proc. Crypt. 1984, Springer-Verlag, 1984, 47-53. doi: 10.1007/3-540-39568-7_5.  Google Scholar

[38]

K. Singh and N. Trichy, Lattice forward-secure identity based encryption scheme, J. Internet Serv. Inf. Sec., 2 (2012), 118-128.   Google Scholar

[39]

Z. WanX. LaiJ. WengS. LiuY. Long and X. Hong, Certificateless key-insulated signature without random oracles, J. Zhejiang Univ. Sci. A, 10 (2009), 1790-1800.  doi: 10.1631/jzus.A0820714.  Google Scholar

[40]

Z. WanX. Meng and X. Hong, Certificateless strong key-insulated signature without random oracles, J. Shanghai Jiaotong Univ. (Sci), 16 (2011), 571-576.  doi: 10.1007/s12204-011-1191-7.  Google Scholar

[41]

B. Waters, Efficient identity-based encryption without random oracles, in Proc. Eurocrypt. 2005, Springer-Verlag, 2005,114-127. doi: 10.1007/11426639_7.  Google Scholar

[42]

J. Weng, X. Li, K. F. Chen and S. L. Liu, Identity-based parallel key-insulated encryption without random oracles, in Proc. Indocrypt. 2006, Springer-Verlag, 2006,409-423. doi: 10.1007/11941378_29.  Google Scholar

[43]

J. Weng, S. L. Liu, K. F. Chen, D. Zheng and W. D. Qiu, Identity-based threshold keyinsulated encryption without random oracles, in Proc. CT-RSA 2008, Springer-Verlag, 2008,203-220. doi: 10.1007/978-3-540-79263-5_13.  Google Scholar

[44]

H. YangS. Sun and H. Li, Forward-secure identity-based encryption scheme (in Chinese), J. Univ. Electr. Sci. Techn. China, 36 (2007), 534-537.   Google Scholar

[45]

D. Yao, N. Fazio, Y. Dodis and A. Lysyanskaya, ID-based encryption for complex hierarchies with applications to forward security and broadcast encryption, in Proc. ACM CCS 2004, ACM, 2004,354-363. doi: 10.1145/1030083.1030130.  Google Scholar

[46]

J. Yu, R. Hao, H. Zhao, M. Shu and J. Fan, IRIBE: Intrusion-resilient identity-based encryption, Inf. Sci. , 329 (2016), 90-104. doi: 10.1016/j.ins.2015.09.020.  Google Scholar

[47]

J. YuF. Y. KongX. G. ChengR. Hao and J. X. Fan, Forward-secure identity-based publickey encryption without random oracles, Fundam. Inf., 111 (2011), 241-256.   Google Scholar

[48]

J. YuF. Y. KongX. G. ChengR. Hao and J. X. Fan, intrusion-resilient identity-based signature: security definition and construction, J. Syst. Softw., 85 (2012), 382-391.  doi: 10.1016/j.jss.2011.08.034.  Google Scholar

[49]

J. Yu, F. Y. Kong, X. G. Cheng, R. Hao and G. W. Li, Construction of yet another forwardsecure signature scheme using bilinear maps, in Proc. ProvSec 2008, Springer-Verlag, 2008, 83-97. doi: 10.1007/978-3-540-88733-1_6.  Google Scholar

Figure 1.  An example of how to associate the time periods $\{0, 1, \dots, 13\}$ with the nodes in a full binary tree with level 3
Figure 2.  An example to show which node secret keys are included in the private key of a user with identity $ID$ in each time period $i (0 \le i \le 13)$
Table 1.  Security of the compared forward-secure identity-based encryption schemes
Compared items Yao et al.'s [45] Yu et al.'s [47] Ours
Standard model? No Yes Yes
Security level $\textit{fs}$-ID-CCA2 $\textit{fs}$-ID-CPA $\textit{fs}$-ID-CCA2
Compared items Yao et al.'s [45] Yu et al.'s [47] Ours
Standard model? No Yes Yes
Security level $\textit{fs}$-ID-CCA2 $\textit{fs}$-ID-CPA $\textit{fs}$-ID-CCA2
Table 2.  Storage costs of the compared forward-secure identity-based encryption schemes
Compared item Yao et al.'s [45] Yu et al.'s [47] Ours
Private key size $O(l')$ $O(l'^2)$ $O(l^2)$
Compared item Yao et al.'s [45] Yu et al.'s [47] Ours
Private key size $O(l')$ $O(l'^2)$ $O(l^2)$
Table 3.  Computation costs of the compared forward-secure identity-based encryption schemes
Compared items Yao et al.'s [45] Yu et al.'s [47] Ours
Key extraction time $O(l')$ $O(l'^2)$ $O(1)$
Key update time $O(l')$ $O(l'^2)$ $O(l)$
Encryption time $O(l')$ $O(l')$ $O(l)$
Decryption time $O(l')$ $O(1)$ $O(1)$
Compared items Yao et al.'s [45] Yu et al.'s [47] Ours
Key extraction time $O(l')$ $O(l'^2)$ $O(1)$
Key update time $O(l')$ $O(l'^2)$ $O(l)$
Encryption time $O(l')$ $O(l')$ $O(l)$
Decryption time $O(l')$ $O(1)$ $O(1)$
Table 4.  Communication costs of the compared forward-secure identity-based encryption
Compared items Yao et al.'s [45] Yu et al.'s [47] Ours
Public parameters size $O(l')$ $O(l')$ $O(l)$
Initial private key size $O(l')$ $O(l'^2)$ $O(1)$
Ciphertext size $O(l')$ $O(1)$ $O(1)$
Compared items Yao et al.'s [45] Yu et al.'s [47] Ours
Public parameters size $O(l')$ $O(l')$ $O(l)$
Initial private key size $O(l')$ $O(l'^2)$ $O(1)$
Ciphertext size $O(l')$ $O(1)$ $O(1)$
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