Another Approach to Pairing Computation in Edwards Coordinates

Ionica, Sorina; Joux, Antoine

doi:10.1007/978-3-540-89754-5_31

Cited by 43 publications

(39 citation statements)

References 19 publications

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“…Several authors have presented new formulas that achieve faster iterations of the Miller loop on certain curves [12,14,27,2,13]. The operation counts presented in these papers are given in the context of Tate pairing computations on curves with even embedding degrees, where all elliptic curve operations occur in the base field F q and the functions in the Miller loop are evaluated at a point which has one coordinate in F q k/2 and one in the full extension field F q k .…”

Section: Computing the Ate Pairing Entirely On The Twisted Curvementioning

confidence: 99%

“…We compare operation counts with the results given by Ionica and Joux [27] and Arène et al [2]; note that those papers cover general Weierstrass curves but we are not aware of any other study covering this case. -(ii) Curves of the form y 2 = x 3 + c 2 have a point of order 3 and admit twists of degrees d = 2 and 6.…”

Section: Comparisonsmentioning

confidence: 99%

“…This started to become more interesting lately for alternative curve shapes such as Edwards curves, studied in [14,27,2], and curves of the form y 2 = x 3 + c 2 , studied in [13].…”

Section: Introductionmentioning

confidence: 99%

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Faster Pairing Computations on Curves with High-Degree Twists

Costello

Lange

Naehrig

2010

Public Key Cryptography – PKC 2010

100

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Abstract. Research on efficient pairing implementation has focussed on reducing the loop length and on using high-degree twists. Existence of twists of degree larger than 2 is a very restrictive criterion but luckily constructions for pairing-friendly elliptic curves with such twists exist. In fact, Freeman, Scott and Teske showed in their overview paper that often the best known methods of constructing pairing-friendly elliptic curves over fields of large prime characteristic produce curves that admit twists of degree 3, 4 or 6.A few papers have presented explicit formulas for the doubling and the addition step in Miller's algorithm, but the optimizations were all done for the Tate pairing with degree-2 twists, so the main usage of the high-degree twists remained incompatible with more efficient formulas.In this paper we present efficient formulas for curves with twists of degree 2, 3, 4 or 6. These formulas are significantly faster than their predecessors. We show how these faster formulas can be applied to Tate and ate pairing variants, thereby speeding up all practical suggestions for efficient pairing implementations over fields of large characteristic.

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Section: Computing the Ate Pairing Entirely On The Twisted Curvementioning

confidence: 99%

Section: Comparisonsmentioning

confidence: 99%

See 1 more Smart Citation

Faster Pairing Computations on Curves with High-Degree Twists

Costello

Lange

Naehrig

2010

Public Key Cryptography – PKC 2010

100

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“…Desde el descubrimiento de las curvas de Edwards en 2007 [3] y de las curvas de Edwards binarias en el 2008 [4], se han presentado algunos resultados tanto en el terreno matemático como en el computacional, que mejoran las condiciones teóricas de esta clase de funciones y su implementación eficiente en diferentes plataformas.…”

Section: Introductionunclassified

Implementation of addition and doubled of a point on a curve in a field Edwards binary // Implementación de suma y doblado de un punto en una curva de Edwards en un campo binario

2017

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RESUMENEn este artículo se presentan los resultados del diseño y desarrollo de operaciones de la aritmética de curvas de Edwards en el campo de Galois GF (2251). Se implementaron las operaciones de suma y doblado de puntos en una curva de Edwards binaria basado en la aritmética de campo finito utilizando una base polinomial. La evaluación de las operaciones se realizó sobre un sistema multiprocesador MPSoC, utilizando las capacidades de multiprocesamiento. En las operaciones de la aritmética en curvas de Edward y campos finitos se utilizan algoritmos eficientes y adecuados para un sistema de multiprocesamiento MPSoC Propeller.Palabras clave: MPSoC; Propeller; Aritmética de curvas de Edwards; Aritmética de campos finitos, Curva binaria de Edwards. ABSTRACTThis article presents the results of the design and development of operations of arithmetic of curves of Edwards in the field of Galois GF (2251). Is implemented the operations of sum and bent of points in a curve of Edwards binary based on the arithmetic of field finite using a base polynomial. The operations were evaluated on a system multiprocessor MPSoC, using multiprocessing capabilities. Operations of arithmetic in finite fields and Edward curves are used in appropriate and efficient algorithms for a multiprocessing system MPSoC Propeller.

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“…Recently, other models, for example, Edwards curves [9,2] and twisted Edwards curves [3] are widely used. Pairing computation on twisted Edward curves was first considered by Das and Sarkar [8] and Ionic and Joux [13]. Then, in 2009, Arène, Lange et al [1] developed explicit formulae for pairing computation on twisted Edwards curves.…”

Section: Introductionmentioning

confidence: 99%

Faster Pairing Computation on Jacobi Quartic Curves with High-Degree Twists

Zhang

2015

Trusted Systems

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Abstract. In this paper, we propose an elaborate geometric approach to explain the group law on Jacobi quartic curves which are seen as the intersection of two quadratic surfaces in space. Using the geometry interpretation we construct the Miller function. Then we present explicit formulae for the addition and doubling steps in Miller's algorithm to compute Tate pairing on Jacobi quartic curves. Both the addition step and doubling step of our formulae for Tate pairing computation on Jacobi curves are faster than previously proposed ones. Finally, we present efficient formulas for Jacobi quartic curves with twists of degree 4 or 6. For twists of degree 4, both the addition steps and doubling steps in our formulas are faster than the fastest result on Weierstrass curves. For twists of degree 6, the addition steps of our formulae are faster than the fastest result on Weierstrass curves.

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Another Approach to Pairing Computation in Edwards Coordinates

Cited by 43 publications

References 19 publications

Faster Pairing Computations on Curves with High-Degree Twists

Faster Pairing Computations on Curves with High-Degree Twists

Implementation of addition and doubled of a point on a curve in a field Edwards binary // Implementación de suma y doblado de un punto en una curva de Edwards en un campo binario

Faster Pairing Computation on Jacobi Quartic Curves with High-Degree Twists

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