The Eta Pairing Revisited

Heß, Florian; Smart, Nigel P.; Vercauteren, Fréderik

doi:10.1109/tit.2006.881709

Cited by 321 publications

(255 citation statements)

References 19 publications

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“…Hence these results do not seem to immediately generalise to any other cases (e.g., [6]), though this is an interesting question for future research.…”

Section: Computing Eta Pairings Without a Final Exponentiationmentioning

confidence: 79%

Simplified pairing computation and security implications

Galbraith¹,

hÉigeartaigh²,

Sheedy³

2007

Journal of Mathematical Cryptology

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Abstract. Recent progress on pairing implementation has made certain pairings extremely simple and fast to compute. Hence, it is natural to examine if there are consequences for the security of pairing-based cryptography. This paper gives a method to compute eta pairings in a way which avoids the requirement for a final exponentiation. The method does not lead to any improvement in the speed of pairing implementation. However, it seems appropriate to re-evaluate the security of pairing based cryptography in light of these new ideas. A multivariate attack on the pairing inversion problem is proposed and analysed. Our findings support the belief that pairing inversion is a hard computational problem.

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“…Hence these results do not seem to immediately generalise to any other cases (e.g., [6]), though this is an interesting question for future research.…”

Section: Computing Eta Pairings Without a Final Exponentiationmentioning

confidence: 79%

Simplified pairing computation and security implications

Galbraith¹,

hÉigeartaigh²,

Sheedy³

2007

Journal of Mathematical Cryptology

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“…For formulae for the higher order twists we refer the reader to [9]. Where the quartic twist applies, #E (F p k/4 ) = q + 1 − f , where f = 4q − τ 2 .…”

Section: Elliptic Curves Over Extension Fieldsmentioning

confidence: 99%

“…For maximum efficiency P and Q are drawn from the groups G 1 of points on E(F p ) and G 2 , a group of points on the twisted curve E (F p d ) where d divides the embedding degree k. For the Tate pairing the first parameter P is chosen from G 1 and the second Q from G 2 . However recent discoveries of the faster ate [9] and R-ate [11] pairings require P to be chosen from G 2 and Q from G 1 . In either case P must be of prime order r, where k, the embedding degree, is the smallest integer for which r|Φ k (t−1) [2], where Φ k (.)…”

Section: Introductionmentioning

confidence: 99%

Fast Hashing to G 2 on Pairing-Friendly Curves

Scott

Benger

Charlemagne

et al. 2009

Lecture Notes in Computer Science

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Abstract. When using pairing-friendly ordinary elliptic curves over prime fields to implement identity-based protocols, there is often a need to hash identities to points on one or both of the two elliptic curve groups of prime order r involved in the pairing. Of these G1 is a group of points on the base field E(F p ) and G 2 is instantiated as a group of points with coordinates on some extension field, over a twisted curve E (F p d ), where d divides the embedding degree k. While hashing to G 1 is relatively easy, hashing to G 2 has been less considered, and is regarded as likely to be more expensive as it appears to require a multiplication by a large cofactor. In this paper we introduce a fast method for this cofactor multiplication on G2 which exploits an efficiently computable homomorphism.

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“…These results were further optimised by Duursma and Lee [15] who developed an inexpensive, closed form for specific parameterisations later improved by Kwon [29]. Their techniques were generalised and extended to produce the Eta [4] and Ate [23] pairings, currently considered the fastest means of evaluation. However, as well as the pairing itself, one depends on lower-level algorithms for arithmetic in the fields F p , F p k/2 and F p k .…”

Section: Introductionmentioning

confidence: 99%

On Software Parallel Implementation of Cryptographic Pairings

Grabher

Großschädl

Page

2009

Selected Areas in Cryptography

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Abstract. A significant amount of research has focused on methods to improve the efficiency of cryptographic pairings; in part this work is motivated by the wide range of applications for such primitives. Although numerous hardware accelerators for pairing evaluation have used parallelism within extension field arithmetic to improve efficiency, similar techniques have not been examined in software thus far. In this paper we focus on parallelism within one pairing evaluation (intra-pairing), and parallelism between different pairing evaluations (inter-pairing). We identify several methods for exploiting such parallelism (extending previous results in the context of ECC) and show that it is possible to accelerate pairing evaluation by a significant factor in comparison to a naive approach.

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The Eta Pairing Revisited

Cited by 321 publications

References 19 publications

Simplified pairing computation and security implications

Simplified pairing computation and security implications

Fast Hashing to G 2 on Pairing-Friendly Curves

On Software Parallel Implementation of Cryptographic Pairings

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