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1624 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 b) (decrypt, ID ∗ , c ∗ ) where c ∗ is the challenge ciphertext. c) Any pair of queries (rkextract, ID ∗ , ID i ), (decrypt, ID i , c i ) where c i =Reencrypt(rk ID∗ →ID i , c ∗ ). In the Guess stage, let ID ∗ be the target i- dentity, and parse the challenge ciphertext c ∗ as (C1 ∗ , C2 ∗ , C3 ∗ , C4 ∗ , C5 ∗ ). In both phases, B responds to A’s queries as follows. • On (extract, ID), where(in the Guess)stage ID ≠ ID ∗ , B selects randomly r i ∈ Zp, ∗ sets sk IDi = (d 0 , d 1 ) = −α ′ H (g 1 (ID i )−T 1 g α′ ) ri , g2 g ri ). We claim sk IDi is a valid random private key b for ID i . To see this, let ˜r i = r i − H . 1(ID i)−T Then we have that 2 (g (H1(IDi)−T ) d 0 = g −α ′ H 1 (ID i )−T 2 (g (H1(IDi)−T ) g2(g a (H1(IDi)−T ) 1 g α′ ) ri− b H 1 (ID i )−T g2(g a H1(IDi) 1 h) ˜ri . −1 H(ID d 1 = g i )−T 2 g ri = g ˜ri . −1 d ′ H(ID 0 = g i )−T 2 g ri = g ˜ri . −1 H 1 (ID i )−T 1 g α′ ) ri = • On (rkextract, ID, ID ′ ), do the same as A handling re-encryption key query in Phase 13 in the above theorem. • On (decrypt, ID, c) where (in the Guess stage) (ID, c) ≠ (ID ∗ , c ∗ ), check whether c is a level-1 (non re-encrypted) or level-2 (reencrypted) ciphertext. In the Guess stage, parse c ∗ as (C1 ∗ , C2 ∗ , C3 ∗ , C4 ∗ , C5 ∗ ). For a level-1 ciphertext, B parses c as (C 1 , C 2 , C 3 , C 4 , C 5 ) and: a) Looks up the value (ID ‖ C 1 ‖ C 2 ‖ C 3 ‖ C 4 ) in the H 4 table, to obtain the tuple (ID ‖ C 1 ‖ C 2 ‖ C 3 ‖ C 4 , z, g z ). If (ID ‖ C 1 ‖ C 2 ‖ C 3 ‖ C 4 ) is not in the table, or if (in the Guess stage) C 5 = C5 ∗ , then B returns ⊥ to A. b) Looks up the value (δ, M, s) in the H 2 table. Checks whether there exist an item of (δ, M, s) such that S = g zs . If not, B returns ⊥ to A. c) Computes k = e(C1,d0) e(C , checks that δ = C 2,d 1) k . If not, B returns ⊥ to A. d) Checks that C 4 = H 3 (δ) ⊕ M. If not, B returns ⊥ to A. e) Otherwise, B returns M to A. For a level-2 ciphertext, B parses c as (C 1, ′ C 2, ′ C 3, ′ C 4, ′ C 5, ′ C 6, ′ C 7, ′ C 8) ′ and: a) Computes k = = C ′ 3e(C ′ 5, C ′ 4) e(C ′ 2 , C′ 6 )e(C′ 1 , C′ 7 )e(d′ 0 , C′ 1 ) C ′ 3e(rk 2 , C ′ 4) e(C ′ 2 , rk 3)e(C ′ 1 , rk 4)e(d ′ 0 , C′ 1 ) = b) Checks that δ = C k . If not, B returns ⊥ to A. c) Checks that C 2 = h s g H1(ID)s 1 . If so, return M. Otherwise, return ⊥. • On (reencrypt, C ID , ID, ID ′ ). B runs ReEnc(rk ID→ID ′, C ID , ID, ID ′ ) and returns the results. At the end of the Find phase, A outputs (ID ∗ , M 0 , M 1 ), with the condition that A has not previously issued (extract, ID ∗ ). At the end of the Guess stage, A outputs its guess bit i ′ . 3) Choice and Challenge. At the end of the Find phase, A outputs (ID ∗ , M 0 , M 1 ). B forms the challenge ciphertext as follows: a) Choose δ ∈ G 1 and p ∈ {0, 1} n randomly, and insert (δ, p) in H 3 table. b) Insert (δ, M b , , g 3 , δ · R, M b ⊕ p) to H 2 table. c) Choose z ∈ Z p randomly, and insert ((g 3 , g3 α′ , δ ·R, M b ⊕p), z, g z ) in the H 4 table. B outputs the challenge ciphertext (C1 ∗ , C2 ∗ , C3 ∗ , C4 ∗ , C5 ∗ ) = (g 3 , g3 α′ , δ · R, M b ⊕ p, g3) z to A and begins the GUESS stage. 4) Forgeries and Abort conditions The adversary may forge C 5 on (C 1 , C 2 , C 3 , C 4 ), but from the security of BLS short signature [7], this probability is negligible. Theorem 5: Suppose the DBDH assumption holds, then our scheme proposed in Section III-F is DGE- IBE-IND-ID-CCA secure for the delegator and proxy’s colluding. Proof: The security proof is same as the above theorem except that it does not allow “A issues up to rekey generation queries on (ID, ID ∗ )”, for B does not know the private key corresponding to ID ∗ . Theorem 6: Suppose the DBDH assumption holds, then our scheme proposed in Section III-F is PKG-OW secure for the delegator, proxy and delegatee’s colluding. Proof: The security proof is same as the proof for Theorem 3. IV. COMPARISON In this section, we give our comparison results with other identity based proxy re-encryption schemes[15], [11], [27], [29]. We compare our schemes with other schemes from two ways. First we concern about schemes’ security, then we concern about schemes’ efficiency. Notations: In Table I, we denote with/without random oracle as W/O RO, assumption as Assum, security model as SecMod, colluding attackers as Colluding, underlying IBE as UnderIBE, stand model as Std, , proxy as P, DGA as delegator, DGE as delegatee. P and DGA means that proxy colludes with delegator, P or DGA means that proxy or delegator is malicious adversary but they never collude. SymEnc-Sec means the security of symmetric encryption. © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1625 TABLE I. IBPRE SECURITY COMPARISON Scheme Security W/O RO Assum SecMod Colluding UnderlyIBE Remark GA07A[15] IND-Pr-ID-CPA RO DBDH Sec.3.1[15] P and DGA BF IBE Weak or P and DGE GA07B[15] IND-Pr-ID-CCA RO DBDH Sec.3.1[15] P and DGA BF IBE Strong or P and DGE M07B [27] IND-Pr-sID-CPA Std DBDH Sec.4.2[27] P or DGA BB 1 IBE Weak or DGE CT07[11] IND-Pr-ID-CPA Std DBDH Sec.4.2[11] P and DGA Waters’ IBE Weak or P and DGE SXC08[29] IND-Pr-ID-CCA Std DBDH Sec.2.6[29] P and DGA Waters’ IBE Strong or P and DGE OursCIII-C IND-Pr-sID-CPA Std DBDH III-B P and DGA Variant of Weak or P and DGE BB 1 IBE OursDIII-F IND-Pr-ID-CCA RO DBDH III-B P and DGA Variant of Strong or P and DGE BB 1 IBE TABLE II. IBPRE EFFICIENCY COMPARISON Scheme Enc Check Reenc Dec Ciph-Len 1stCiph 2-ndCiph 1stCiph 2-ndCiph GA07A[15] 1t e + 1t p 0 1t p 2t p 1t p 2|G| + 2|G e| 1|G| + 1|G e| GA07B[15] 1t p + 1t e 2t p 2t e + 2t p 1t e + 2t p 2t e + 2t p 1|G| + 1|G e| 1|G| + 1|G T | +2|m| + |id| +1|G e| + |m| M07B [27] 1t p + 2t e 2t p 1t p 2t p 2t p 2|G e| + 1|G T | 2|G e| + 1|G T | CT07[11] 3t e + 1t p + 1t s 1t v 2t e 2t e + 10t p + 1t v 2t e + 3t p 9|G| + 2|G T | 3|G| + |G T | +|vk| + |s| +|vk| + |s| SXC08[29] 3t e + 1t p + 1t s 1t v 2t e + 1t s 2t e + 10t p + 2t v 2t e + 3t p + 1t v 9|G| + 2|G T | 3|G| + |G T | +2|vk| + 2|s| +1|vk| + 1|s| OursCIII-C 2t e + 1t p 2t p 1t e 4t p 2t p 6|G| + |G T | 2|G| + |G T | OursDIII-F 3t e + 1t me 2t p 1t e 4t p + 1t e + 1t me 2t p + 1t e + 1t me 7|G| + m 4|G| + m From Table I, we can know that our IBPRE scheme based on a variant of BB 1 IBE scheme is the most secure IBPRE. M07B scheme is the weakest IBPRE for it can only achieve IND-Pr-sID-CPA under separated proxy or delegator or delegatee attack. In Table II, we denote encryption as Enc, reencryption as Reenc, decryption as Dec, ciphertext as Ciph and ciphertext length as Ciph-Len. t p , t e and t me represent the computational cost of a bilinear pairing, an exponentiation and a multi-exponentiation respectively, while t s and t v represent the computational cost of a one-time signature signing and verification respectively. |G|, |Z q |, |G e | and |G T | denote the bit -length of an element in groups G, Z q , G e and G T respectively. Here G and Z q denote the groups used in our scheme, while G e and G T are the bilinear groups used in GA07, CT07, SXC08 schemes, i.e., the bilinear pairing is e : G e × G e → G T . Finally, |vk| and |s| denote the bit length of the one-time signature’s public key and a one-time signature respectively. From Table II, Our schemes 3 , GA07 4 and M07B schemes are much more efficient than CT07 and SXC08 scheme due to their underlying IBE is Waters’ IBE. And for the proxy, CT07 and SXC08 scheme are much 3 Our first level ciphertext maps second level ciphertext and second level ciphertext maps first level ciphertext in [15], [11], [29]. Sometimes in our schemes we use e : G × G → G 1 or e : G 1 × G 1 → G T , in the former cases, G maps to G e, G 1 maps G T , in the latter case, G 1 maps to G e, G T maps G T . 4 GA07 and SXC08 are multi-hop IBPRE but we just consider their single-hop variant. more efficient than others for their special paradigm, our IBPRE scheme is more efficient than GA07B scheme and our other schemes, we think this is important for resisting DDos attack against the proxy. V. CONCLUSIONS AND OPEN PROBLEMS In 2007, Matsuo proposed the concept of four types of PRE schemes: CBE to CBE, IBE to CBE, CBE to IBE and IBE to IBE [27]. In Matsuo’s scheme, they allow the PKG to help the delegator and the delegatee to generate re-encryption key. We explore this feature further, if we allow PKG to generate re-encryption keys by directly using master − key, many open problems can be solved. Considering the standardization of BB 1 IBE and its broad applications, we give new identity based proxy re-encryption schemes based on BB 1 IBE, and prove its security in our new stronger security models. Furthermore, our schemes are very efficient for the reencryption process, which is the most heavy-load part of PRE. ACKNOWLEDGEMENT The authors would like to thank Dr. Jian Weng, Dr. Jun Shao, Dr. Licheng Wang, Dr. Fagen Li, Dr. Qiang Tang for many helpful discussions and the anonymous referees for helpful comments. © 2013 ACADEMY PUBLISHER
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Page 261: Call for Papers and Special Issues
Page 264: (Contents Continued from Back Cover
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