A New Method of Building More Non-supersingular Elliptic Curves

Cui, Shi; Duan, Pu; Chan, Choong Wah

doi:10.1007/11424826_69

Cited by 2 publications

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References 11 publications

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“…Recent efforts (cf. [7], [10]) have focused on reducing the ratio ρ = log q/ log r for arbitrary k, but no additional examples have been found with ρ small enough to allow for curves of prime order.…”

Section: Introductionmentioning

confidence: 99%

Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10

Freeman

2006

Lecture Notes in Computer Science

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We present a general framework for constructing families of elliptic curves of prime order with prescribed embedding degree. We demonstrate this method by constructing curves with embedding degree k = 10, which solves an open problem posed by Boneh, Lynn, and Shacham [6]. We show that our framework incorporates existing constructions for k = 3, 4, 6, and 12, and we give evidence that the method is unlikely to produce infinite families of curves with embedding degree k > 12.

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Section: Introductionmentioning

confidence: 99%

Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10

Freeman

2006

Lecture Notes in Computer Science

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“…From the above three equations, Cocks and Pinch [29] In [44], we observed that (1 − t) k ≡ 1 (mod r) is correct when k is even, and concluded that (1 − t) is a primitive k-th root of unity modulo r when k is even. However, this conclusion is not precise.…”

Section: A Methods For Building Non-supersingular Elliptic Curvesmentioning

confidence: 92%

Efficient group signature and identity-based signature schemes

Cui¹

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This thesis introduces a group signature scheme based on RSA and some identitybased signature schemes based on the bilinear pairings over elliptic curves. A practical group signature scheme is proposed with the introduction of a trusted third party called security mediator. The length of group signatures generated by the proposed scheme is shorter at the security level of 1024 bits than Boneh-Boyen-Shacham scheme. It is suitable for those applications which require short signatures. The implementation of the new group signature scheme is very simple and just requires modulo exponentiations. A new identity-based signature (IBS) schemes is developed which are proven secure in the random oracle model. The security of IBS using random oracles relies on Generalized k-CAA which is an extension version of k-CAA. The experimental results show that the proposed IBS scheme in the random oracle model is the fastest one compared to the published IBS schemes from the bilinear pairings at low security levels. We also investigate methods of building non-supersingular elliptic curves suitable for pairing-based cryptosystems and find a useful property of the Frobenius trace for such curves. Based on this property, some non-supersingular elliptic curves can be built. Batch verification is a very efficient method for speeding up multiple signa-ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library iii ture verification. We investigate batch verification of the proposed IBS scheme in the random oracle model. The experimental results exhibit excellent performance in terms of the time for verification of multiple signatures and the implementation complexity. The security analysis shows that batch verification of the proposed IBS scheme is unforgeable under an adaptive chosen message and identity attack. Based on the proposed IBS scheme in the random oracle model, two new signature schemes with special features are developed: identity-based blind signature (IBBS) scheme and identity-based chameleon signature (IBCS) scheme. The security analysis illustrates that the new IBBS and IBCS are provably secure. The performance comparison shows that the efficiency of both new signature schemes is improved compared to the Zhang-Kim IBBS scheme and Zhang-Naini-Susilo IBCS scheme.

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A New Method of Building More Non-supersingular Elliptic Curves

Cited by 2 publications

References 11 publications

Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10

Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10

Efficient group signature and identity-based signature schemes

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