Identity-Based Encryption Protocols Using Bilinear Pairing

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So, the challenge ciphertext is ( CT = M b × e(P 1 , P 2 ) γ , γP, γ ( u ′ )) ∑ V j . CT is a valid encryption of M b under v ∗ = (v1, ∗ . . . , vu ∗ ′). On the other hand, when T is random, CT is random from the view point of A. j=1 Phase 2: 2 together. This is similar to Phase 1. Note that A places at most q queries in Phase 1 and Guess: Finally, A outputs its guess b ′ ∈ {0, 1}. B outputs 1 ⊕ b ⊕ b ′ . A’s view in the above simulation is identical to that in a real attack. This gives us the required bound on the advantage of the adversary in breaking the HIBE protocol. 9.5 Composite Scheme We have mentioned in Section 3.4 that Boneh-Boyen-Goh [19] proposed a “product” construction based on the BBG-HIBE and the BB-HIBE. A similar construction is possible based on the HIBE G 1 of Section 9.3 and BB-HIBE. The resulting HIBE is secure in s + ID model. On the other hand, in Chapter 7 we have presented a construction H 1 which is secure in model M 1 . This construction is in a sense an extension of the BB-HIBE. We propose a composite scheme based on H 1 and G 2 which we denote as (h, n)-G 3 or simply G 3 . The essential idea, as in [19] is to form a product of two HIBEs. For this we represent an identity tuple in the form of a matrix (say II) having (a-priori) fixed number of columns, h. When we look at a row of II, it forms a constant ciphertext HIBE, H, while each row taken together as a single identity forms another HIBE, H ′ . We obtain a product construction by instantiating H ′ to be H 1 of Chapter 7 and H to be the constant size ciphertext HIBE G 2 of Section 9.4. In this case, the components of the identity tuples are from Z p and we obtain security in M 1 . Since M 1 allows the target identity to be of any length up to the maximum height of the HIBE, the adversary has the flexibility to choose the length the target identity in the challenge phase. 9.5.1 Construction Let the maximum depth of the HIBE be h ≤ l 1 × l 2 . Here individual identity components are elements of Z p . 120
Setup: Let P be a generator of G 1 . Choose a random secret x ∈ Z p and set P 1 = xP . Randomly choose P 2 ; an l 1 × l 2 matrix R where ⎡ ⎤ R 1,1 · · · R 1,l2 ⎢ ⎥ R = ⎣ . . . ⎦ R l1 ,1 · · · R l1 ,l 2 and l 2 many vectors −→ U 1 , . . . , −→ U l2 from G 1 , where each −→ U i = (U i,1 , . . . , U i,n ). The public parameters are 〈P, P 1 , P 2 , R, −→ U 1 , . . . , −→ U l2 〉, while the master secret is xP 2 . Key Generation: Given an identity v = (v 1 , . . . , v u ) for any u, this algorithm generates the private key d v of v as follows. Let u = k 1 l 2 + k 2 with k 2 ∈ {1, . . . , l 2 }. We represent v by a (possibly incomplete) (k 1 + 1) × l 2 matrix I where the last row has k 2 elements. Choose (k 1 + 1) many random elements r 1 , . . . , r k1 , r k2 ∈ Z p and output ( ( ∑k 1 h∑ ) ( k2 ) ∑ d ID = xP 2 + r i V i,j + R i,j + r k2 V k1 +1,j + R k1 +1,j , r 1 P, . . . , r k1 P, r k2 P, i=1 j=1 j=1 ) −−−→ −→ r k2 R k1 +1,k 2 +1, . . . , r k2 R k1 +1,h, r k2 U k2 +1, . . . , r k2Ul2 where V i,j = ∑ n −→ j=1 vj U i,j and r k2Ui denotes that each element of −→ U i is multiplied by the scalar r k2 . The private key of v can also be generated given the private key of v |u−1 = v 1 , . . . , v u−1 as required. There are two cases to be considered. Case 1: Suppose u − 1 = k 1 l 2 + l 2 , then ( ( k∑ 1 +1 h∑ ) ) d v|u−1 = xP 2 + r i V i,j + R i,j , r 1 P, . . . , r k1 P, r k1 +1P i=1 j=1 = (a 0 , a 1 , . . . , a k1 , a k1 +1) (say) Choose a random r ∗ ∈ Z p and form d v as d v = a 0 + r ∗ (V k1 +2,1 + R k1 +2,1), a 1 , . . . , a k1 +1, r ∗ P, r ∗ R k1 +2,2, . . . , r ∗ R k1 +2,h, r ∗ −→ U2 , . . . , r ∗ −→ Ul2 Case 2: Let, u − 1 = k 1 l 2 + k 2 ′ with k 2 ′ < l 2 then, ( ∑k 1 h∑ ) ⎛ ⎞ k ∑2 ′ d v|u−1 = (xP 2 + r i V i,j + R i,j + r k ′ ⎝ 2 V k1 ,j + P k1 ⎠ +1,j , r 1 P, . . . , r k1 P, r k ′ 2 P, i=1 j=1 j=1 r k ′ 2 R k1 +1,k 2 ′ +1 , . . . , r k ′ 2 R k1 +1,h, r k ′ −→ 2 U k ′ 2 +1, . . . , r k ′ −→ 2 U l2 = (a 0 , a 1 , . . . , a k1 , a k1 +1, b k ′ 2 +1, . . . , b l2 , −→ c k ′ 2 +1, . . . , −→ c l2 ) (say) 121
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Construction of (Hierarchical) Iden
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The scientist does not study nature
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Contents 1 Introduction 1 1.1 Plan
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7.4.1 HIBE H 1 . . . . . . . . . .
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the public key. In this setting, an
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adversary in distinguishing two mes
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previous chapter. The second varian
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as it is closer to the known exampl
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Let P i = a i P . An algorithm A ha
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Given the security parameter κ, th
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the Key-Gen algorithm and C is the
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Phase 2: A now issues additional qu
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Chapter 3 Previous Works in Identit
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Decrypt: Decrypt C = 〈U, V 〉 us
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Game 2 Suppose B is given a BDH pro
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Security of BasicHIBE against chose
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BB-HIBE Here individual components
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F i (v ∗ i ) = α i P , so C =
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Key-Gen: Let v = (v 1 , . . . , v n
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Composite HIBE Boneh, Boyen and Goh
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Security Given an identity tuple v
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Chapter 4 Tate Pairing in General C
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order on the elliptic curve E(IF p
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the best result. In [57] the author
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Hence, we require 3[S] + 8[M] for E
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Algorithm 2 (iterated doubling): In
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Level 1 : t 1 = Y 2 1 ; t 4 = Z 2 1
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Table 4.1: Cost Comparison. Note 1[
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problem. - Our construction resembl
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is equal to c 2 |IF a | 3 . Thus, t
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aborts if F (v ∗ ) ≠ 0 and outp
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DBDH problem over (G 1 , G 2 , e) i
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these are respectively 2.4 kb and 2
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Signing: Let M = (m 1 , m 2 , . . .
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model without random oracles. Addit
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6.2 Construction HIBE-spp The ident
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Table 6.1: Comparison of HIBE Proto
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For 1 ≤ j ≤ h, we define severa
Page 82 and 83: establish the claim. We provide the
Page 84 and 85: The probability is over the indepen
Page 86 and 87: = ≥ ( ( 1 − Pr 1 − [( q∨ i=
Page 88 and 89: of public parameters is significant
Page 90 and 91: to be I ∗ . It can then ask for t
Page 92 and 93: Phase 2: The adversary issues addit
Page 94 and 95: 7.3 Interpreting Security Models Th
Page 96 and 97: 7.4 Constructions We present severa
Page 98 and 99: 7.4.2 HIBE H 2 The description of H
Page 100 and 101: Since b 0,1 , . . . , b 0,h , b 1 ,
Page 102 and 103: For 1 ≤ i ≤ h, we have V i (y)
Page 104 and 105: 1. The encryption is converted into
Page 106 and 107: Chapter 8 Constant Size Ciphertext
Page 108 and 109: Let a private key for (v 1 , . . .
Page 110 and 111: Phase 1: Suppose a key extraction q
Page 112 and 113: = γ ( P 3 + ) u∑ V i . i=1 If th
Page 114 and 115: Encrypt: To encrypt a message M ∈
Page 116 and 117: and outputs a random bit if there i
Page 118 and 119: Chapter 9 Augmented Selective-ID Mo
Page 120 and 121: Phase 1 and Phase 2: As discussed i
Page 122 and 123: an adversary has to commit to an id
Page 124 and 125: Table 9.1: Comparison of BBG-HIBE a
Page 126 and 127: where ˜r = (r − αk F k ). Also
Page 128 and 129: The maximum height of the HIBE be h
Page 130 and 131: B declares the public parameters to
Page 134 and 135: Choose a random r ∗ ∈ Z p and f
Page 136 and 137: The size of the private key in the
Page 138 and 139: generalisation and move on to addre
Page 140 and 141: Table 10.3: Comparison of HIBE prot
Page 142 and 143: [9] Paulo S. L. M. Barreto, Ben Lyn
Page 144 and 145: [31] Sanjit Chatterjee and Palash S
Page 146 and 147: [55] Florian Hess, Nigel P. Smart,
Page 148: [81] Palash Sarkar. Head: Hybrid en
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Identity-Based Encryption Protocols Using Bilinear Pairing

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