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1620 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 and delegatee security. Delegator Security. In PRE from IBE to IBE, we consider the case that proxy and delegatee are corrupted. Definition 3: (DGA-IBE-IND-ID-CPA) A PRE scheme from IBE to IBE is DGA 1 -IBE-IND-ID-CPA secure if the probability P r[{(ID ⋆ , sk ID ⋆) ← KeyGen(·)} {(ID x , sk IDx ) ← KeyGen(·)}, {(ID h , sk IDh ) ← KeyGen(·)}, {R hx ← RKGen(msk, sk IDh , sk IDx , ·)}, {R xh ← RKGen(msk, sk IDx , sk IDh , ·)}, {R hh ← RKGen(msk, sk IDh , sk IDh , ·)}, {R xx ← RKGen(msk, sk IDx , sk IDx , ·)}, {R ⋆h ← RKGen(msk, sk ID ⋆, sk IDh , ·)}, {R ⋆x ← RKGen(msk, sk ID ⋆, sk IDx , ·)}, (m 0 , m 1 , St) ← A Orenc (ID ⋆ , {sk IDx }, {R xh }, {R hx }, {R hh }, {R xx }, {R ⋆h }, {R ⋆x }), d ⋆ R ←− {0, 1}, C ⋆ = Encrypt(m d ⋆, ID ⋆ ), d ′ ← A Ørenc (C ⋆ , St) : d ′ = d ⋆ ] is negligibly close to 1/2 for any PPT adversary A. In our notation, St is a state information maintained by A while (ID ⋆ , sk ID ⋆) is the target user’s pubic and private key pair generated by the challenger which also chooses other keys for corrupt and honest parties. For other honest parties, keys are subscripted by h and we subscript corrupt keys by x. Oracles O renc proceeds as follows: • Re-encryption O renc : on input (pk i , ID j , C pki ), where C pki is the ciphertext under the public key pk i , pk i were produced by Keygen CBE , ID j were produced by Keygen IBE , this oracle responds with ‘invalid’ if C pki is not properly shaped w.r.t. pk i . Otherwise the re-encrypted first level ciphertext C ID = ReEnc(KeyGen P RO (sk i , ID j , mk, parms), ID j , parms, C pki ) is returned to A. Delegatee Security. In PRE from IBE to IBE, we consider the case that proxy and delegator are corrupted. Definition 4: (DGE-IBE-IND-ID-CPA) A PRE scheme from IBE to IBE is DGE 2 -IBE-IND-ID-CPA 1 DGA means Delegator 2 DGE means Delegatee. secure if the probability P r[{(ID ⋆ , sk ID ⋆) ← KeyGen(·)} {(ID x , sk IDx ) ← KeyGen(·)}, {(ID h , sk IDh ) ← KeyGen(·)}, {R hx ← RKGen(msk, sk IDh , sk IDx , ·)}, {R xh ← RKGen(msk, sk IDx , sk IDh , ·)}, {R hh ← RKGen(msk, sk IDh , sk IDh , ·)}, {R xx ← RKGen(msk, sk IDx , sk IDx , ·)}, {R h⋆ ← RKGen(msk, sk IDh , sk ID ⋆, ·)}, {R x⋆ ← RKGen(msk, sk IDx , sk ID ⋆, ·)}, (m 0 , m 1 , St) ← A Orenc (ID ⋆ , {sk IDx }, {R xh }, {R hx }, {R hh }, {R xx }, {R h⋆ }, {R x⋆ }), d ⋆ R ←− {0, 1}, C ⋆ = Encrypt(m d ⋆, ID ⋆ ), d ′ ← A Ørenc (C ⋆ , St) : d ′ = d ⋆ ] is negligibly close to 1/2 for any PPT adversary A. The notations in this game are same as Definition 3. PKG Security. In PRE from IBE and IBE, PKG’s master key can not leverage even if the delegator, the delegatee and proxy collude. Definition 5: (PKG-OW) A PRE scheme from IBE to IBE is one way secure for PKG if the probability P r[{(ID x , sk IDx ) ← KeyGen(·)}, {(ID h , sk IDh ) ← KeyGen(·)}, {R hx ← RKGen(msk, sk IDh , sk IDx , ·)}, {R xh ← RKGen(msk, sk IDx , sk IDh , ·)}, {R hh ← RKGen(msk, sk IDh , sk IDh , ·)}, {R xx ← RKGen(msk, sk IDx , sk IDx , ·)}, mk ′ ← A Orenc ({sk IDx }, {sk IDh }, {R xh }, {R hx }, {R hh }, {R xx }, {parms}) : mk = mk ′ ] is negligibly close to 0 for any PPT adversary A. The notations in this game are same as Definition 3. C. Our Proposed IND-Pr-sID-CPA Secure IBPRE Scheme Based on a Variant of BB 1 IBE • The underlying IBE scheme: We give a variant of BB 1 -IBE scheme as follows: Let G be a bilinear group of prime order p (the security parameter determines the size of G). Let e : G × G → G 1 be the bilinear map. For now, we assume public keys (ID) is element in Zp. ∗ We later extend the construction to public keys over {0, 1} ∗ by first hashing ID using a collision resistant hash H : {0, 1} ∗ → Z p . We also assume messages to be encrypted are elements in G. The IBE system works as follows: 1) SetUp IBE (k). Given a security parameter k, select a random generator g ∈ G and random elements g 2 = g t1 , h = g t2 ∈ G. Pick a random α ∈ Zp. ∗ Set g 1 = g α ,mk = g2 α , and params = © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1621 (g, g 1 , g 2 , h). Let mk be the master-secret key and let params be the public parameters. KeyGen IBE (mk, params, ID). Given mk = g2 α and ID with params, the PKG picks random s 0 , s 1 ∈ Zp, ∗ choose a hash function ˜H : Zp ∗ × {0, 1} ∗ → Zp ∗ and computes u 0 = ˜H(s0 , ID), u 1 = ˜H(s 1 , ID). Set sk ID = (d 0 , d 1 , d ′ 0) = (g2 α (g1 ID h) u0 , g u0 , (g2 α (g1 ID h) u1 )). The PKG preserves (s 0 , s 1 ). Enc IBE (ID, params, M). To encrypt a message M ∈ G 1 under the public key ID ∈ Zp, ∗ pick a random r ∈ Zp ∗ and compute C ID = (g r , (g1 ID h) r , Me(g 1 , g 2 ) r ). Dec IBE (sk ID , params, C ID ). Given ciphertext C ID = (C 1 , C 2 , C 3 ) and the secret key sk ID = (d 0 , d 1 ) with prams, compute M = C 3e(d 1,C 2) e(d . 0,C 1) delegation scheme: KeyGen PRO (sk R , params, ID, ID ′ ). The PKG computes u ′ 1 = ˜H(s 1 , ID ′ ) and randomly selects k 1 , k 2 , k 3 ∈ Zp ∗ and sets rk ID→ID ′ = (rk 1 , rk 2 , rk 3 , rk 4 ) = ( αID′ +t 2+k 1 k 3(αID+t 2) + k 2 , g u′ 1 k3 , g u′ 1 k2k3 , g u′ 1 k1 ) and sends them to the proxy via secure channel. We must note that the PKG computes a different (k 1 , k 2 , k 3 ) for every different user pair (ID, ID ′ ). Check(params, C ID , ID). Given the delegator’s identity ID and C ID = (C 1 , C 2 , C 3 ) with params, compute v 0 = e(C 1 , g1 ID h) and v 1 = e(C 2 , g). If v 0 = v 1 then output 1. Otherwise output 0. ReEnc(rk ID→ID ′, params, C ID , ID ′ ). Given the identities ID, ID ′ , rk ID→ID ′ = (rk 1 , rk 2 , rk 3 , rk 4 ) = ( αID′ +t 2+k 1 k 3(αID+t 2) + k 2 , g u′ 1 k3 , g u′ 1 k2k3 , g u′ 1 k1 ) with params, the proxy re-encrypt the ciphertext C ID into C ID ′ as follows. First it runs “Check”, if output 0, then return “Reject”. Else computes C 2ID ′ = (C 1, ′ C 2, ′ C 3, ′ C 4, ′ C 5, ′ C 6, ′ C 7) ′ = αID ′ +t 2 +k 1 k (C 1 , C 2 , C 3 , C (αID+t 2 ) +k2 2 , rk 2 , rk 3 , rk 4 ). Dec1 IBE (sk ID ′, params, C 2ID ′). Given a re-encrypted ciphertext C 2ID ′ = (C 1, ′ C 2, ′ C 3, ′ C 4, ′ C 5, ′ C 6, ′ C 7) ′ and the secret key sk ID = (d 0 , d 1 , d ′ 0) with params, computes C M = 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 ) Dec2 IBE (sk ID ′, params, C 1ID ′). Given a normal ciphertext C ID ′ = (C 1 , C 2 , C 3 ) and the secret key sk ID ′ = (d 0 , d 1 , d ′ 0) with prams, compute M = C3e(d1,C2) e(d . 0,C 1) We Remark computes pair +t 2+k 3(αID+t same not secure Security Theorem our IND-sID-CPA colluding. Proof: construct On output = g interacting Initialization. with intends Setup.To rithm h params ing g2 a Phase • • can verify its correctness as following C 3e(rk ′ 2 , C 4) ′ e(C 2 ′ , rk 3)e(C 1 ′ , rk 4)e(d ′ 0 , C′ 1 ) Me(g 1 , g 2 ) r e(g k3u′ 1 , (g ID = 1 h) r( αID′ +t 2 +k 1 k 3 (αID+t 2 ) +k2) ) e((g1 IDh)r , g u′ 1 k2k3 )e(g r , g k1u′ 1)e(g2 α(gID′ 1 h) u′ 1, g r ) = Me(g 1, g 2 ) r e(g k3u′ 1 , (g ID 1 h) k2r )e(g k3u′ 1 , (g ID ′ 1 h) r e((g1 IDh)r , g u′ 1 k2k3 )e(g r , g k1u′ 1)e(g2 α(gID′ = Me(g 1, g 2 ) r e(g2 α, = M gr ) 2: In our scheme, we must note that the P- a different (k 1 , k 2 , k 3 ) for every different (ID, ID ′ ). Otherwise, if the adversary knows 1 2) 2 for five different pairs (ID, ID ′ ) but k 1 , k 2 , k 3 , α, t 2 , he can compute (α, t 2 ), which at all. Analysis 1: Suppose the DBDH assumption holds, scheme proposed in Section III-C is DGA-IBEsecure for the proxy and the delegatee’s Suppose A can attack our scheme, we an algorithm B solves the DBDH problem in input (g, g a , g a2 , g b , g c , T ), algorithm B’s goal 1 if T = e(g, g) abc and 0 otherwise. Let , g 2 = g b , g 3 = g c . Algorithm B works by with A in a selective identity game as follows: The selective identity game begins A first outputting an identity ID ∗ that it to attack. generate the system’s parameters, algo- B picks α ′ ∈ Z p at random and defines = g1 −ID∗ g α′ ∈ G. It gives A the parameters = (g, g 1 , g 2 , h). Note that the correspond- master − key, which is unknown to B, is = g ab ∈ G ∗ . 1 “A issues up to private key queries on ID i ”. B selects randomly r i , r ′ ∗ i ∈ Z p and k ′ ∈ Z p , sets sk IDi = (d 0 , d 1 , d ′ 0) = −α ′ ID (g i −ID ∗ 2 (g (IDi−ID∗ ) 1 g a ) ri −1 ID , g i −ID ∗ 2 g ri , −α ′ ID g i −ID ∗ 2 (g (IDi−ID∗ ) 1 g a ) r′ i). We claim sk IDi is a valid random private key for ID i . b To see this, let ˜r i = r i − ID−ID and ∗ ˜r i ′ = r′ i − b ID−ID . Then we have that ∗ −α ′ ID d 0 = g i −ID ∗ 2 (g (IDi−ID∗ ) 1 g α′ ) ri = g2(g a (IDi−ID∗ ) 1 g α′ ) ri− b ID−ID∗ = g2(g a IDi 1 h) ˜ri . −1 ID d 1 = g i −ID ∗ 2 g ri = g ˜ri . −α ′ d ′ ID 0 = g i −ID ∗ 2 (g (IDi−ID∗ ) 1 g α′ ) r′ i = g2(g a (IDi−ID∗ ) 1 g α′ ) r′ i − b ID−ID∗ = g2(g a IDi 1 h) ˜r i ′ . “A issues up to rekey generation queries on (ID, ID ′ )”. The challenge B chooses a randomly x ∈ Zp, ∗ 2) KG 3) αID ′ k the is 4) D. • The then 1) G. is to g a 1 1) 2) 2) 3) 3) 4) 5) k 3 )e(g k3u′ 1 , g k 1 r 1 h) u′ 1, g r ) k 3 ) © 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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