Identity-Based Encryption Protocols Using Bilinear Pairing

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establish the claim. We provide the details. Let Y i be the event that the simulator outputs 1 in Game i, i = 1, 2. Then, we have |Pr[Y 1 ] − Pr[Y 2 ]| ≤ ɛ dbdh . Let ab i be the event that the simulator aborts in Game i, i = 1, 2. This includes both protocol and artificial abort. Following the analysis of [89] and [77], we have λ − λɛ 2 ≤ Pr[ab i|X i ], Pr[ab i |X i ] ≤ λ + λɛ 2 . (6.3.5) Here ɛ = ɛ hibe and λ is the lower bound on the probability of not abort up to the artificial abort stage (see Section 6.3.2). Pr[Y i ] = Pr[Y i ∧ (ab i ∨ ab i )] = Pr[(Y i ∧ ab i ) ∨ (Y i ∧ ab i )] = Pr[Y i ∧ ab i ] + Pr[Y i ∧ ab i ] = Pr[Y i | ab i ]Pr[ab i ] + Pr[Y i | ab i ]Pr[ab i ] = 1 2 (1 − Pr[ab i]) + Pr[X i | ab i ]Pr[ab i ] = 1 2 (1 − Pr[ab i ∧ (X i ∨ X i )]) + Pr[X i ∧ ab i ] = 1 2 + 1 2 ( Pr[abi |X i ]Pr[X i ] − Pr[ab i |X i ]Pr[X i ] ) Now we need to do some manipulations with inequalities and for convenience we set A i = Pr[ab i |X i ], B i = Pr[X i ] and C i = Pr[ab i |X i ] and D = Pr[Y 1 ] − Pr[Y 2 ]. We have from (6.3.5) Also λ − λɛ 2 ≤ A i, C i ≤ λ + λɛ 2 . 2D = (A 1 B 1 − C 1 (1 − B 1 )) − (A 2 B 2 − C 2 (1 − B 2 )). (6.3.6) Since both B 1 and (1 − B 1 ) are non-negative, we have Hence, B i (λ − λɛ) ≤ A 2 iB i ≤ B i (λ + λɛ) 2 (1 − B i )(−λ − λɛ) ≤ −C 2 i(1 − B i ) ≤ (1 − B i )(−λ + λɛ). 2 λ(2B i − 1) − λɛ 2 ≤ A iB i − C i (1 − B i ) ≤ λ(2B i − 1) + λɛ 2 . (6.3.7) Putting i = 1 in (6.3.7), we obtain λ(2B 1 − 1) − λɛ 2 ≤ A 1B 1 − C 1 (1 − B 1 ) ≤ λ(2B 1 − 1) + λɛ 2 . (6.3.8) 70
Multiplying (6.3.7) by −1 and putting i = 2 we obtain −λ(2B 2 − 1) − λɛ 2 ≤ −(A 2B 2 − C 2 (1 − B 2 )) ≤ −λ(2B 2 − 1) + λɛ 2 . (6.3.9) Combining (6.3.6), (6.3.8) and (6.3.9) we get 2λ(B 1 − B 2 ) − λɛ ≤ 2D ≤ 2λ(B 1 − B 2 ) + λɛ. (6.3.10) This shows that |λ(B 1 − B 2 ) − D| ≤ λɛ. Now |λ(B 2 1 − B 2 )| − |D| ≤ |λ(B 1 − B 2 ) − D| ≤ λɛ Note that |D| = |Pr[Y 1 ] − Pr[Y 2 ]| ≤ ɛ dbdh and recalling the values of B 1 and B 2 , we have |Pr[X 1 ] − Pr[X 2 ]| ≤ ɛ dbdh λ + ɛ hibe 2 . This completes the proof of the claim. Now we can complete the proof in the following manner. ɛ hibe = ∣ Pr[X 0] − 1 2∣ ≤ |Pr[X 0 ] − Pr[X 2 ]| ≤ |Pr[X 0 ] − Pr[X 1 ]| + |Pr[X 1 ] − Pr[X 2 ]| ≤ ɛ hibe 2 + ɛ dbdh λ . Rearranging the inequality gives the desired result. This completes the proof. 2 . 6.3.2 Lower Bound on Not Abort We require the following two independence results in obtaining the required lower bound. Similar independence results have been used earlier in [89, 77] in connection with IBE protocols. The situation for HIBE is more complicated than IBE and especially so since we reuse some of the public parameters over different levels of the HIBE. This makes the proofs more difficult. Our independence results are given in Proposition 6.3.1. and 6.3.2. and these subsume the results for the IBE of previous chapter. We provide complete proofs for these two propositions as well as a complete proof for the lower bound. The probability calculation for the lower bound is also more complicated compared to the IBE case. Proposition 6.3.1. Let m be a prime and L(·) be as defined in (6.3.3). Let v 1 , . . . , v j be identities, i.e., each v i = (v i,1 , . . . , v i,l ), with v i,k to be an n/l-bit string (and hence 0 ≤ v i,k ≤ 2 n/l − 1). Then [ j∧ Pr (L k (v k ) = 0) k=1 ] = 1 m j . 71
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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
Page 32 and 33: Decrypt: Decrypt C = 〈U, V 〉 us
Page 34 and 35: Game 2 Suppose B is given a BDH pro
Page 36 and 37: Security of BasicHIBE against chose
Page 38 and 39: BB-HIBE Here individual components
Page 40 and 41: F i (v ∗ i ) = α i P , so C =
Page 42 and 43: Key-Gen: Let v = (v 1 , . . . , v n
Page 44 and 45: Composite HIBE Boneh, Boyen and Goh
Page 46 and 47: Security Given an identity tuple v
Page 48 and 49: Chapter 4 Tate Pairing in General C
Page 50 and 51: order on the elliptic curve E(IF p
Page 52 and 53: 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 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
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So, the challenge ciphertext is ( C
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Choose a random r ∗ ∈ Z p and f
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The size of the private key in the
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generalisation and move on to addre
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Table 10.3: Comparison of HIBE prot
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[9] Paulo S. L. M. Barreto, Ben Lyn
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[31] Sanjit Chatterjee and Palash S
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[55] Florian Hess, Nigel P. Smart,
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[81] Palash Sarkar. Head: Hybrid en
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Identity-Based Encryption Protocols Using Bilinear Pairing

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