Identity-Based Encryption Protocols Using Bilinear Pairing

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Table 9.1: Comparison of BBG-HIBE and G 1 . Protocol/ id PP max pvt dec. subkey/ enc. eff. dec. eff. security Sec. Model comp size key size CT expansion degradation G 1 in s + ID Z p 3 + 2h 2h 2 h + 2 2 none BBG in s + ID Z ∗ p 4 + h h + 1 2 h + 2 2 h BBG in sID Z ∗ p 4 + h h + 1 2 h + 2 2 none The columns PP size, max pvt key size, dec. subkey/CT expansion stand for the number of elements of G 1 in public parameter, private key, decryption subkey and ciphertext expansion respectively. The column enc. eff stands for the number of scalar multiplications in G 1 during encryption while the column dec. eff denotes the number of pairing computations during decryption. All values are for a HIBE of maximum height h. Comparison to BBG protocol The protocol G 1 is a modification of the BBG-HIBE with a different P 3,i for each level of the HIBE. This is required to get a proof of security in the augmented s + -ID model without any security degradation. This reduction is provided in the next section. Additionally, it allows identity components to be elements of Z p , instead of Z ∗ p as in BBG-HIBE. On the other hand, this modification only affects the efficiency of the BBG-HIBE in a small way. The first thing to note is the size of the ciphertext is still constant (three elements). Secondly, the size of the public parameter as well as private key is linear in the length of the HIBE and private key size decreases as we “go down” the HIBE. These two properties ensure that the applications mentioned in [19] also hold for the new HIBE described above. In particular, it is possible to combine the new HIBE with the BB-HIBE of [17] to get an intermediate HIBE with controllable trade-off between the size of the ciphertext and the size of the private key. Further, the application to the construction of forward secure encryption protocol mentioned in [19] can also be done with the new HIBE. 9.3.2 Security Semantic security (i.e., (CPA-security) of the above scheme in the s + -ID model is proved under the h-wDBDHI ∗ assumption. Theorem 9.3.1. Let (t ′ , ɛ, h)-wDBDHI ∗ assumption holds in 〈G 1 , G 2 , e()〉. Then the HIBE G 1 is (t, q, ɛ)-IND-sID-CPA secure for q ≥ 1, t ′ ≈ t + O(τq) where τ is the time for a scalar multiplication in G 1 . Proof : Suppose A is a (t, q, ɛ)-CPA adversary for the G 1 , then we construct an algorithm B that solves the h-wDBDHI ∗ problem. B takes as input a tuple 〈P, Q, Y 1 , . . . , Y h , T 〉 where 112
Y i = α i P for some random α ∈ Z ∗ p and T is either equal to e(P, Q) αh+1 or a random element of G 2 . We define the s + ID game between B and A as follows. Initialization: A outputs an identity tuple v ∗ = (v1, ∗ . . . , vu) ∗ ∈ (Z p ) u for any u ≤ h. The restriction on A is that it cannot ask for the private key of v ∗ or any of its prefixes and in challenge stage it asks for an encryption under v ∗ or any of its prefixes. In case u < h, B chooses random vu+1, ∗ . . . , vh ∗ from Z p and keeps these extra elements to itself. (Note that B is not augmenting the target identity to create a new target identity.) Setup: B picks random β, β 1 , . . . , β h and c 1 , . . . , c h in Z p . It then sets P 1 = Y 1 = αP ; P 2 = Y h + βP = (α h + β)P ; and for 1 ≤ j ≤ u, Q j = β j P − Y h−j+1 ; P 3,j = c j P + v ∗ jY h−j+1 ; and for u < j ≤ h, Q j = β j P ; P 3,j = c j P + v ∗ jY h−j+1 . B declares the public parameter as (P, P 1 , P 2 , −→ P 3 , −→ Q), where −→ Q = (Q 1 , . . . , Q h ), −→ P 3 = (P 3,1 , . . . , P 3,h ). The corresponding master key αP 2 = Y h+1 +βY 1 is unknown to B. B defines the functions F j = v ∗ j − v j for 1 ≤ j ≤ u and F j = v ∗ j for u < j ≤ h and J j = c j + β j v j for 1 ≤ j ≤ h. Phase 1: Suppose A asks for the private key corresponding to an identity v = (v 1 , . . . , v m ) for m ≤ h. Note that for any j ≤ u, Similarly, for u < j ≤ h V j = P 3,j + v j Q j = c j P + vjY ∗ h−j+1 + v j (β j P − Y h−j+1 ) = (v ∗ j − v j )Y h−j+1 + (c j + β j v j )P = F j Y h−j+1 + J j P. V j = P 3,j + v j Q j = c j P + v ∗ jY h−j+1 + v j β j P = F j Y h−j+1 + J j P. Hence, V j for 1 ≤ j ≤ h is computable from what is known to B. Recall that u is the length of v ∗ that the adversary committed to before the set-up phase. If m ≤ u, then there must be a k ≤ τ such that F k ≠ 0, as otherwise the queried identity is a prefix of the target identity. In case m > u, it is possible that F 1 = · · · = F u = 0. Then by construction, F u+1 ≠ 0. Let k be the smallest in {1, . . . , m} such that F k ≠ 0. B picks a random r ∈ Z p and assigns d 0|k = (−J k /F k )Y k + βY 1 + rV k and d 1 = (−1/F k )Y k + rP. Now, d 0|k = − J k F k Y k + βY 1 + α k Y h−k+1 − α k F k F k Y h−k+1 + rV k = αP 2 + ˜rV k 113
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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 , . . .
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 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 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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