Identity-Based Encryption Protocols Using Bilinear Pairing

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previous chapter. The second variant (called FullccHIBE) is secure in the full model. Like BBG-HIBE, both are proved secure without random oracle. In Chapter 9, we augment the selective-ID security model for HIBE by allowing the adversary some flexibility in choosing the target identity tuple during the challenge phase of the security reduction. We denote this model by selective + -ID model (s + ID model). We show that the HIBE proposed by Boneh-Boyen [17] satisfies this notion of security. The case of the constant size ciphertext HIBE of Boneh, Boyen and Goh is different – the original reduction in [19] can be modified to achieve this goal where security degrades by a multiplicative factor of the number of levels in the HIBE. Further we modify the BBG-HIBE to construct a new protocol called G 1 , secure in s + ID model without any degradation. G 1 maintains all the attractive features of BBG-HIBE. We build on this new construction another constant size ciphertext HIBE G 2 . The security of G 2 is proved under the augmented version of M 2 . Our third construction in this chapter is a “product” HIBE G 3 that allows a controllable trade-off between the ciphertext size and the private key size. We conclude the dissertation by summing-up our contributions in Chapter 10. We also mention some open problems in the area. 6
Chapter 2 Preliminaries In this chapter, we define the basic notions used throughout the dissertation. The identitybased encryption protocols that we describe all use cryptographic bilinear map as the primary building block. Hence, we start with a definition of bilinear map. 2.1 <strong>Bilinear</strong> Map Let G 1 and G 2 be two cyclic groups of same prime order p for some large p where we write G 1 additively and G 2 multiplicatively. Let P be a generator of G 1 , i.e., G 1 = 〈P 〉. A mapping e : G 1 × G 1 → G 2 is called an admissible bilinear map if it satisfies the following properties: • <strong>Bilinear</strong>ity: e(aQ, bR) = e(Q, R) ab for all Q, R ∈ G 1 and a, b ∈ Z p . • Non-degeneracy: If G 1 = 〈P 〉, then G 2 = 〈e(P, P )〉. • Computability: There exist efficient algorithms to compute the group operations in G 1 , G 2 as well as the map e(). Let, Q = xP and R = yP ; then e(Q, R) = e(P, P ) xy = e(yP, xP ) = e(Q, R). So the map e() also satisfies the symmetry property. Known examples of e() usually have G 1 to be a group of Elliptic Curve (EC) or Hyper Elliptic Curve points and G 2 to be a subgroup of a multiplicative group of a finite field. Modified Weil pairing [20], Tate pairing [8, 45], Eta pairing [7], Ate pairing [55] are examples of bilinear maps. In mathematical literature, it is usual to write elliptic curve group operation additively whereas it is usual to write the finite field group operation multiplicatively. Consequently, in works on pairing computation and implementation of pairing based protocols [8, 45, 11], it is customary to write G 1 additively and G 2 multiplicatively. Initial works on protocol design such as [20, 49] also adhered to this notation. However, later works on protocol design [17, 18, 89] write both G 1 and G 2 multiplicatively. Here we follow the first convention 7
Page 1: Construction of (Hierarchical) Iden
Page 5: The scientist does not study nature
Page 9 and 10: Contents 1 Introduction 1 1.1 Plan
Page 11: 7.4.1 HIBE H 1 . . . . . . . . . .
Page 14 and 15: the public key. In this setting, an
Page 16 and 17: adversary in distinguishing two mes
Page 20 and 21: as it is closer to the known exampl
Page 22 and 23: Let P i = a i P . An algorithm A ha
Page 24 and 25: Given the security parameter κ, th
Page 26 and 27: the Key-Gen algorithm and C is the
Page 28 and 29: Phase 2: A now issues additional qu
Page 30 and 31: 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
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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
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establish the claim. We provide the
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The probability is over the indepen
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= ≥ ( ( 1 − Pr 1 − [( q∨ i=
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of public parameters is significant
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to be I ∗ . It can then ask for t
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Phase 2: The adversary issues addit
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7.3 Interpreting Security Models Th
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7.4 Constructions We present severa
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7.4.2 HIBE H 2 The description of H
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Since b 0,1 , . . . , b 0,h , b 1 ,
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For 1 ≤ i ≤ h, we have V i (y)
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1. The encryption is converted into
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Chapter 8 Constant Size Ciphertext
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Let a private key for (v 1 , . . .
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Phase 1: Suppose a key extraction q
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= γ ( P 3 + ) u∑ V i . i=1 If th
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Encrypt: To encrypt a message M ∈
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and outputs a random bit if there i
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Chapter 9 Augmented Selective-ID Mo
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Phase 1 and Phase 2: As discussed i
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an adversary has to commit to an id
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Table 9.1: Comparison of BBG-HIBE a
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where ˜r = (r − αk F k ). Also
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The maximum height of the HIBE be h
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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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