Identity-Based Encryption Protocols Using Bilinear Pairing

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1. The encryption is converted into a hybrid scheme. Instead of multiplying the message with the “mask” Z = e(P 1 , P 2 ) s , the value Z is provided as input to a pseudorandom generator and the message (considered to be a bit string) is XORed with the resulting keystream. 2. The private key corresponding to an identity v is d v = (xP 2 + rV v , rP ), where V v = P 3,1 + V (v) as defined in Section 7.4.1. 3. Suppose the intended set of receivers is {v 1 , . . . , v j }. Then the ciphertext consists of the encryption of the message plus a header of the form (sP, sV 1 , . . . , sV j ), where V i is as defined in the construction of H 1 in Section 7.4.1 and s is a random element of Z p . 4. The receiver possessing the secret key d vi (1 ≤ i ≤ j) can compute e(P 1 , P 2 ) s in the standard manner and hence obtain the input to the pseudorandom generator. Thus it can decrypt the message. The MR-IBE described above can be proved to be secure in the selective-ID model without random oracle. The security model for MR-IBE is the following [4]. In the commitment stage, the adversary commits to a set of identities; does not ask for the private key of these identities in the key extraction queries and finally asks for the encryption under this set of identities. Note that this is very similar to the model M 1 restricted to IBE. The only difference is that during the generation of the challenge ciphertext, in M 1 , the adversary supplies only one identity out of the set of identities it had previously committed to, whereas in the model for MR-IBE, the adversary asks for the encryption under the whole set of these identities. This difference is easily tackled in our proof in Section 7.5.1 which shows that H 1 is secure in model M 1 . Recall that the construction of the polynomial F (x) is such that F (v) = 0 for all v ∈ I ∗ , where I ∗ is the set of committed identities. In the challenge stage of the security proof for H 1 as an IBE, we use this fact for only one identity (the identity given by the adversary). In the proof for MR-IBE, we will need to generate cV i for all v ∈ I ∗ . Since F (v) = 0 for any such v, this can be done in the standard fashion. The above argument does not provide any security degradation. Hence, we obtain an MR-IBE which can be proved to be secure in the selective-ID model without random oracle. 7.7 Conclusion In this chapter, we have generalized the notion of selective-ID secure HIBE. Two new security models M 1 and M 2 have been introduced. In the security game, both these models allow an adversary to commit to a set of identities (as opposed to a single identity in the sID model) before the set-up. During the challenge stage, the adversary can choose any one of the previously committed identities as a challenge identity. We provide two HIBE constructions H 1 and H 2 which are secure in the models M 1 and M 2 respectively. The public parameter size is smaller in case of H 1 . Further, we also show that H 1 and H 2 can be modified to 92
obtain an MR-IBE protocol which is secure in the sID model without random oracles. The only previous construction of MR-IBE is secure in the sID model using random oracle. 93
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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
Page 54 and 55: Hence, we require 3[S] + 8[M] for E
Page 56 and 57: Algorithm 2 (iterated doubling): In
Page 58 and 59: Level 1 : t 1 = Y 2 1 ; t 4 = Z 2 1
Page 60 and 61: Table 4.1: Cost Comparison. Note 1[
Page 62 and 63: problem. - Our construction resembl
Page 64 and 65: 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 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 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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