Identity-Based Encryption Protocols Using Bilinear Pairing

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and outputs a random bit if there is no such j. Otherwise, it answers the query in a manner similar to that in the proof of Theorem 8.2.1.. B chooses r randomly from Z p and computes d 0|j = − J(v j) F j (v j ) Y j + yY 1 + r(F j (v j )Y h−j+1 + J(v j )P ); d 1 = −1 F j (v j ) Y j + rP. It is standard to show that d 0|j = αP 2 +˜rV j and d 1 = ˜rP , where ˜r = r− αj F j (I j . As in the proof ) of Theorem 8.2.1., it is possible to show that B can compute ˜rV i for any i ∈ {1, . . . , u} \ {j}; and ˜rP 3,k , ˜r −→ U k for u < k ≤ h. The simulator computes d 0 = d 0|j + ∑ i∈{1,...,u}\{j} ˜rV i. A is provided the private key corresponding to v as d v = (d 0 , d 1 , ˜rP 3,u+1 , . . . , ˜rP 3,h , ˜r −→ U u+1 , . . . , ˜r −→ ) U h . Note that d v is a valid private key for v following the proper distribution. B will be able to generate this d v as long as there is a j ∈ {1, . . . , u} such that F (v j , k j ) ≢ 0 for which it suffices to have K(v j ) ≠ 0. Challenge: A submits two messages M 0 , M 1 ∈ G 2 and an identity v ∗ = (v1, ∗ . . . , vh ∗ ′), h ′ ≤ h on which it wants to be challenged. B aborts and outputs a random bit, if F j (vj) ∗ ≢ 0 for any j ∈ {1, . . . , h ′ }. Otherwise, B chooses a random bit γ ∈ {0, 1} and gives A the tuple CT = (T × e(Y 1 , yQ) × M γ , Q, ∑ h ′ j=1 J(v∗ j)Q). If 〈P, Q, Y 1 , . . . , Y h , T 〉 given to B is a valid h-wDBDHI ∗ tuple, i.e., T = e(P, Q) αh+1 then CT is a valid encryption for M γ . Suppose Q = cP for some unknown c ∈ Z p . Then the first component of CT can be seen to be e(P 1 , P 2 ) c . Further, using F j (vj) ∗ ≡ 0 mod p it can be shown that J(vj)Q ∗ = cV j . The correctness of the third component of CT follows from this fact. If T is a random element of G 2 , CT gives no information about B’s choice of γ. Phase 2: Similar to Phase 1, with the restriction that A cannot ask for the private key of ID ∗ or any of its ancestors. Guess: A outputs a guess γ ′ of γ. A lower bound λ on the probability of aborting up to this stage is the following. λ = 1 2(4lq2 n/l ) h . Waters [89] obtains a similar bound (for the case l = n) in the context of an IBE secure in the full model under the DBDH assumption. In the same paper, Waters had suggested a construction for HIBE where new public parameters are generated for each level of the HIBE. Generating new public parameters for each level of the HIBE simplifies the probability analysis for the lower bound on the probability of abort. 104
The security of FullccHIBE is based on the h-wDBDHI ∗ problem which is obtained by modifying the BBG-HIBE. For each level of the HIBE we have separate public parameters P 3,i and (U i,1 , . . . , U i,l ). This makes it possible to easily apply the reasoning of Waters leading to the above mentioned lower bound. At this point, we have to also use the technique of “artificial abort” employed by Waters [89]. The idea is that the probability of aborting up to this is not independent of the adversarial queries. The idea of the artificial abort technique is to allow the simulator to sample the transcript of queries it obtained from the adversary and on certain conditions abort and output a random bit. This increases the total probability of abort and makes it almost equal for all adversarial inputs. This helps in the probability analysis. The effect of sampling the transcript is to increase the runtime of the simulator. See [89] for the details. Incorporating the probability analysis of Waters into the present situation in a straightforward manner we obtain the required result. 8.4 Conclusion In this chapter, we have presented two variants of the constant size ciphertext HIBE proposed by Boneh, Boyen and Goh [19]. Both the constructions have constant size ciphertext expansion. The first variant is proved to be secure in the generalized selective-ID model M 2 introduced in Chapter 7 while the second variant is secure in the full model. We combine techniques from several works along with the BBG-HIBE to obtain the new constructions and their proofs. 105
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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
Page 66 and 67: aborts if F (v ∗ ) ≠ 0 and outp
Page 68 and 69: DBDH problem over (G 1 , G 2 , e) i
Page 70 and 71: these are respectively 2.4 kb and 2
Page 72 and 73: Signing: Let M = (m 1 , m 2 , . . .
Page 74 and 75: model without random oracles. Addit
Page 76 and 77: 6.2 Construction HIBE-spp The ident
Page 78 and 79: Table 6.1: Comparison of HIBE Proto
Page 80 and 81: 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 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 132 and 133: So, the challenge ciphertext is ( C
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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