Identity-Based Encryption Protocols Using Bilinear Pairing

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Table 4.1: Cost Comparison. Note 1[I]≥ 30[M] [72]. Method BKLS [8] (affine) Izu-Takagi [57] (general a) Scott [82] (a = −3) Algorithm 1 (a small) Algorithm 1 (a = −3) Algorithm 2 (general a) Cost t(1[I]+8[M]+2[S])+s(1[I]+6[M]+1[S]) t(19[M]+9[S])+s(22[M]+4[S]) 21t[M]+22s[M] t(12[M]+6[S])+s(14[M]+3[S]) t(13[M]+4[S])+s(14[M]+3[S]) t(13[M]+5[S])+s(14[M]+4[S]) We consider the cost comparison to EC scalar multiplication. For the purpose of security, scalar multiplication has to be resistant to side channel attacks. One simple method of attaining resistance to simple power analysis is to use Coron’s dummy addition using binary representation of multiplier. Under the (realistic) assumption that the length of the binary representation of the multiplier is equal to the length of the NAF representation of r for Tate pairing, the cost of dummy-addition countermeasure is t(2[M]+7[S]) for the case of a = −3. This cost is comparable to the cost of Algorithm 1 for a = −3 when s is at most around t/8. Again for practical situation r can usually be chosen so that s ≤ t/8. Thus the efficiency of our algorithm is almost comparable to the efficiency of simple SPA resistant EC scalar multiplication. On the other hand, there is a wide variety of techniques for EC scalar multiplication providing very efficient algorithms. Whether the cost of Tate pairing computation can be made comparable to the most efficient EC scalar multiplication is currently a challenging research problem. 4.6 Conclusion In this chapter, we have considered the use of Jacobian coordinates for Tate pairing computation in general characteristics with embedding degree two. The main idea that we have introduced is encapsulated double-and-line computation and encapsulated add-and-line computation. We have also developed encapsulated version of iterated double algorithm. The algorithms are presented in details and memory requirement has been considered. Inherent parallelism in these algorithms have been identified leading to optimal two-multiplier parallel algorithms. Our algorithms lead to an improvement of around 33% over previously best known algorithm for the general case where the curve parameter a is an arbitrary field element. In the special case where a = −3, our techniques provide an efficiency improvement of around 20% over the previously best known algorithm. 48
Chapter 5 Identity-Based Encryption in Full Model 5.1 Introduction In this chapter, we provide a generalisation of the identity-based encryption scheme of Waters [89]. As already noted in Chapter 3, one disadvantage of Waters’ scheme in [89] is the requirement of a rather large public parameter file. If identities are represented by a bit string of length n, then the scheme requires an n length vector of elements of G 1 to be maintained as part of the public parameter. Our generalisation shows that if one tries to reduce the number of elements in the public parameter then there is a corresponding degradation in the security reduction. In other words, a trade-off is involved in the tightness of security reduction and smallness of public parameter. The trade-off between tightness and smallness can be converted to a trade-off between group size and smallness of public parameter. When desiring a specific security level, the loss of security due to loss of tightness in the security reduction can be compensated by working in a larger group. This increases the bit length of representation of the elements in the public parameter but the number of elements in the public parameters decreases so drastically that there is a significant reduction in the overall size of the public parameter. The increase in group size in turn affects the efficiency of the protocol. Thus, the trade-off is actually between the space required to store the parameters and the time required to execute the protocol. For example, if identities are represented by 160-bit strings, then Waters protocol require to store 160 elements of G 1 as part of the public parameter. Alternatively, using our generalisation if one wants to store 16 elements, then to achieve 80-bit security, compared to Waters protocol the space requirement reduces by around 90% while the computation cost increases by around 30%. – Like Waters, applying Naor’s technique, our scheme can also be easily converted to a signature scheme where the underlying security assumption is the computational Diffie-Hellman 49
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 18 and 19: previous chapter. The second varian
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 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 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
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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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