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116
Efficient and generalized pairing computation on Abelian varieties
, 2008
"... In this paper, we propose a new method for constructing a bilinear pairing over (hyper)elliptic curves, which we call the Rate pairing. This pairing is a generalization of the Ate and Atei pairing, and also improves efficiency of the pairing computation. Using the Rate pairing, the loop length in ..."
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Cited by 54 (3 self)
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In this paper, we propose a new method for constructing a bilinear pairing over (hyper)elliptic curves, which we call the Rate pairing. This pairing is a generalization of the Ate and Atei pairing, and also improves efficiency of the pairing computation. Using the Rate pairing, the loop length in Miller’s algorithm can be as small as log(r 1/φ(k) ) for some pairingfriendly elliptic curves which have not reached this lower bound. Therefore we obtain from 29 % to 69 % savings in overall costs compared to the Atei pairing. On supersingular hyperelliptic curves of genus 2, we show that this approach makes the loop length in Miller’s algorithm shorter than that of the Ate pairing.
Optimal Pairings
"... Abstract. In this paper we introduce the concept of an optimal pairing, which by definition can be computed using only log 2 r/ϕ(k) basic Miller iterations, with r the order of the groups involved and k the embedding degree. We describe an algorithm to construct optimal ate pairings on all parametri ..."
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Cited by 51 (0 self)
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Abstract. In this paper we introduce the concept of an optimal pairing, which by definition can be computed using only log 2 r/ϕ(k) basic Miller iterations, with r the order of the groups involved and k the embedding degree. We describe an algorithm to construct optimal ate pairings on all parametrized families of pairing friendly elliptic curves. Finally, we conjecture that any nondegenerate pairing on an elliptic curve without efficiently computable endomorphisms different from powers of Frobenius requires at least log 2 r/ϕ(k) basic Miller iterations.
Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves
"... Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the GallantLambertVanstone (GLV) method. Iijima, Matsuo, Chao and Tsujii gave such homomorphisms for a large class of elliptic curves by working over F p 2. We extend their results and demonstra ..."
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Cited by 49 (3 self)
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Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the GallantLambertVanstone (GLV) method. Iijima, Matsuo, Chao and Tsujii gave such homomorphisms for a large class of elliptic curves by working over F p 2. We extend their results and demonstrate that they can be applied to the GLV method. In general we expect our method to require about 0.75 the time of previous best methods (except for subfield curves, for which Frobenius expansions can be used). We give detailed implementation results which show that the method runs in between 0.70 and 0.83 the time of the previous best methods for elliptic curve point multiplication on general curves.
NanoECC: Testing the limits of elliptic curve cryptography in sensor networks
 Proceedings of the 5th European conference on Wireless Sensor Networks, LNCS 4913
, 2008
"... Abstract. By using Elliptic Curve Cryptography (ECC), it has been recently shown that PublicKey Cryptography (PKC) is indeed feasible on resourceconstrained nodes. This feasibility, however, does not necessarily mean attractiveness, as the obtained results are still not satisfactory enough. In thi ..."
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Cited by 46 (4 self)
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Abstract. By using Elliptic Curve Cryptography (ECC), it has been recently shown that PublicKey Cryptography (PKC) is indeed feasible on resourceconstrained nodes. This feasibility, however, does not necessarily mean attractiveness, as the obtained results are still not satisfactory enough. In this paper, we present results on implementing ECC, as well as the related emerging field of PairingBased Cryptography (PBC), on two of the most popular sensor nodes. By doing that, we show that PKC is not only viable, but in fact attractive for WSNs. As far as we know pairing computations presented in this paper are the most efficient results on the MICA2 (8bit/7.3828MHz ATmega128L) and Tmote Sky (16bit/8.192MHz MSP430) nodes.
Progressionfree sets and sublinear pairingbased noninteractive zeroknowledge arguments
 In TCC
, 2012
"... Abstract. In 2010, Groth constructed the only previously known sublinearcommunication NIZK circuit satisfiability argument in the common reference string model. We optimize Groth’s argument by, in particular, reducing both the CRS length and the prover’s computational complexity from quadratic to q ..."
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Cited by 42 (3 self)
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Abstract. In 2010, Groth constructed the only previously known sublinearcommunication NIZK circuit satisfiability argument in the common reference string model. We optimize Groth’s argument by, in particular, reducing both the CRS length and the prover’s computational complexity from quadratic to quasilinear in the circuit size. We also use a (presumably) weaker security assumption, and have tighter security reductions. Our main contribution is to show that the complexity of Groth’s basic arguments is dominated by the quadratic number of monomials in certain polynomials. We collapse the number of monomials to quasilinear by using a recent construction of progressionfree sets.
Faster explicit formulas for computing pairings over ordinary curves
, 2010
"... We describe e cient formulas for computing pairings on ordinary elliptic curves over prime fields. First, we generalize lazy reduction techniques, previously considered only for arithmetic in quadratic extensions, to the whole pairing computation, including towering and curve arithmetic. Second, we ..."
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Cited by 38 (8 self)
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We describe e cient formulas for computing pairings on ordinary elliptic curves over prime fields. First, we generalize lazy reduction techniques, previously considered only for arithmetic in quadratic extensions, to the whole pairing computation, including towering and curve arithmetic. Second, we introduce a new compressed squaring formula for cyclotomic subgroups and a new technique to avoid performing an inversion in the final exponentiation when the curve is parameterized by a negative integer. The techniques are illustrated in the context of pairing computation over BarretoNaehrig curves, where they have a particularly efficient realization, and also combined with other important developments in the recent literature. The resulting formulas reduce the number of required operations and, consequently, execution time, improving on the stateoftheart performance of cryptographic pairings by 27%33% on several popular 64bit computing platforms. In particular, our techniques allow to compute a pairing under 2 million cycles for the first time on such architectures.
ON CRYPTOGRAPHIC PROTOCOLS EMPLOYING ASYMMETRIC PAIRINGS – THE ROLE OF Ψ REVISITED
"... Abstract. Asymmetric pairings e: G1 × G2 → GT for which an efficientlycomputable isomorphism ψ: G2 → G1 is known are called Type 2 pairings; if such an isomorphism ψ is not known then e is called a Type 3 pairing. Many cryptographic protocols in the asymmetric setting rely on the existence of ψ for ..."
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Cited by 27 (3 self)
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Abstract. Asymmetric pairings e: G1 × G2 → GT for which an efficientlycomputable isomorphism ψ: G2 → G1 is known are called Type 2 pairings; if such an isomorphism ψ is not known then e is called a Type 3 pairing. Many cryptographic protocols in the asymmetric setting rely on the existence of ψ for their security reduction while some use it in the protocol itself. For these reasons, it is believed that some of these protocols cannot be implemented with Type 3 pairings, while for some the security reductions either cannot be transformed to the Type 3 setting or else require a stronger complexity assumption. Contrary to these widely held beliefs, we argue that Type 2 pairings are merely inefficient implementations of Type 3 pairings, and appear to offer no benefit for protocols based on asymmetric pairings from the point of view of functionality, security, and performance. 1.
An introduction to pairingbased cryptography
, 2005
"... Bilinear pairings have been used to design ingenious protocols for such tasks as oneround threeparty key agreement, identitybased encryption, and aggregate signatures. Suitable bilinear pairings can be constructed from ..."
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Cited by 26 (0 self)
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Bilinear pairings have been used to design ingenious protocols for such tasks as oneround threeparty key agreement, identitybased encryption, and aggregate signatures. Suitable bilinear pairings can be constructed from
Pairing Lattices
 In Pairing 2009, volume 5209 of Lecture
"... Abstract. We provide a convenient mathematical framework that essentially encompasses all known pairing functions based on the Tate pairing and also applies to the Weil pairing. We prove nondegeneracy and bounds on the lowest possible degree of these pairing functions and show how endomorphisms can ..."
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Cited by 25 (1 self)
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Abstract. We provide a convenient mathematical framework that essentially encompasses all known pairing functions based on the Tate pairing and also applies to the Weil pairing. We prove nondegeneracy and bounds on the lowest possible degree of these pairing functions and show how endomorphisms can be used to achieve a further degree reduction. 1
Highspeed software implementation of the optimal ate pairing over Barreto–Naehrig curves
 PAIRINGBASED CRYPTOGRAPHY–PAIRING 2010. LECTURE NOTES IN COMPUTER SCIENCE
, 2010
"... This paper describes the design of a fast software library for the computation of the optimal ate pairing on a Barreto–Naehrig elliptic curve. Our library is able to compute the optimal ate pairing over a 254bit prime field Fp, injust2.33 million of clock cycles on a single core of an Intel Core ..."
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Cited by 25 (3 self)
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This paper describes the design of a fast software library for the computation of the optimal ate pairing on a Barreto–Naehrig elliptic curve. Our library is able to compute the optimal ate pairing over a 254bit prime field Fp, injust2.33 million of clock cycles on a single core of an Intel Core i7 2.8GHz processor, which implies that the pairing computation takes 0.832msec. We are able to achieve this performance by a careful implementation of the base field arithmetic through the usage of the customary Montgomery multiplier for prime fields. The prime field is constructed via the Barreto–Naehrig polynomial parametrization of the prime p given as, p =36t 4 +36t 3 +24t 2 +6t +1, with t =2 62 − 2 54 +2 44. This selection of t allows us to obtain important savings for both the Miller loop as well as the final exponentiation steps of the optimal ate pairing.