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Faster computation of the Tate pairing

Christophe Arène, Tanja Lange, Michael Naehrig, Christophe Ritzenthaler
2011 Journal of Number Theory  
The Tate pairing on Edwards curves can be computed by using these functions in Miller's algorithm.  ...  This paper proposes new explicit formulas for the doubling and addition steps in Miller's algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form.  ...  Ionica and Joux [23] use a different map to a curve of degree 3 and compute the 4-th power of the Tate pairing.  ... 
doi:10.1016/j.jnt.2010.05.013 fatcat:gbw7ngc6unbblaqygwkbalw4re

Faster computation of the Tate pairing [article]

Christophe Arene, Tanja Lange, Michael Naehrig, Christophe Ritzenthaler
2010 arXiv   pre-print
This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing.  ...  Computing the coefficients of the functions and the sum or double of the points is faster than with all previously proposed formulas for pairings on Edwards curves.  ...  Ionica and Joux [23] use a different map to a curve of degree 3 and compute the 4-th power of the Tate pairing.  ... 
arXiv:0904.0854v3 fatcat:utbwvxy4inghphsllhnwn2gnhi

TinyTate: Computing the Tate Pairing in Resource-Constrained Sensor Nodes

Leonardo B. Oliveira, Diego F. Aranha, Eduardo Morais, Felipe Daguano, Julio Lopez, Ricardo Dahab
2007 Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007)  
On the other hand, the advent of cryptography from pairings has enabled a wide range of novel cryptosystems.  ...  In this work we present TinyTate, the first known implementation of pairings for sensor nodes based on the 8bit/7.3828-MHz ATmega128L microcontroller (e.g., MICA2 and MICAz motes).  ...  Figure 1 . 1 Tate pairing computation: Miller's algorithm and final exponentiation. Figure 2 . 2 Tate pairing computation using the BKLS algorithm. .21s.  ... 
doi:10.1109/nca.2007.48 dblp:conf/nca/OliveiraAMDLD07 fatcat:gs6anqesdzhireq5hgh32chrja

Refinements of Miller's algorithm for computing the Weil/Tate pairing

Ian F. Blake, V. Kumar Murty, Guangwu Xu
2006 Journal of Algorithms  
In this paper we propose three refinements to Miller's algorithm for computing Weil/Tate Pairing.  ...  We also have some performance analysis. keywords: algorithm, elliptic curve, cryptography, Weil/Tate pairing Recent work on parings in cryptography has considered their use in the definition and implementation  ...  The function φ n is called Tate pairing. An essential part in computing the Weil/Tate pairing is the evaluation of f P (R) for each point R in the support of D Q .  ... 
doi:10.1016/j.jalgor.2005.01.009 fatcat:sahhckccpvc5rpgzkaii7ddb3a

Efficient Computations of the Tate Pairing for the Large MOV Degrees [chapter]

Tetsuya Izu, Tsuyoshi Takagi
2003 Lecture Notes in Computer Science  
The computation time of the Tate pairing strongly depends on underlying elliptic curves and definition fields.  ...  We develop several formulas that compute the Tate pairing using the small number of multiplications over IF q k .  ...  Acknowledge The authors would like to thank anonymous refrees for their helpful comments and suggestions.  ... 
doi:10.1007/3-540-36552-4_20 fatcat:w4ztie4kazglxfswg2ntu5wjfq

Computing the Cassels-Tate Pairing in the Case of a Richelot Isogeny [article]

Jiali Yan
2021 arXiv   pre-print
We also include a worked example demonstrating we can turn the descent by Richelot isogeny into a 2-descent via computing the Cassels-Tate pairing.  ...  We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on Sel^ϕ(J) ×Sel^ϕ(J) where ϕ is the dual isogeny of ϕ.  ...  In [FN14] , Fisher and Newton computed the Cassels-Tate pairing on Sel 3 (E) × Sel 3 (E).  ... 
arXiv:2109.08257v1 fatcat:urdp3r7vhrdf7kn6qaxht25cgy

The yoga of the Cassels–Tate pairing

Tom Fisher, Edward F. Schaefer, Michael Stoll
2010 LMS Journal of Computation and Mathematics  
AbstractCassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels–Tate pairing.  ...  In this article, we prove that the two pairings are the same.  ...  The authors are grateful for the helpful comments of three anonymous referees.  ... 
doi:10.1112/s1461157010000185 fatcat:3jofamomvzb4tgk7n4tqlmohkm

Implementing the Tate Pairing [chapter]

Steven D. Galbraith, Keith Harrison, David Soldera
2002 Lecture Notes in Computer Science  
To efficiently implement these cryptosystems it is necessary to optimise the computation time for the Tate pairing. This paper provides methods to achieve fast computation of the Tate pairing.  ...  Tate pairing, bilinear pairing, implementing characteristic 3 The Weil and Tate pairings have found several new applications in cryptography.  ... 
doi:10.1007/3-540-45455-1_26 fatcat:6ny5mblrdjcdfgbymgtlcf56my

TATE PAIRING COMPUTATION ON THE DIVISORS OF HYPERELLIPTIC CURVES OF GENUS 2

Eun-Jeong Lee, Yoon-Jin Lee
2008 Journal of the Korean Mathematical Society  
Furthermore, we obtain a very efficient algorithm for the Tate pairing computation over divisors by reducing the cost of computing.  ...  We find a completely general method for computing the Tate-pairing over divisor class groups of the curves H b in a very explicit way.  ...  total number of iterations for the Tate pairing computation over such curves.  ... 
doi:10.4134/jkms.2008.45.4.1057 fatcat:uennwyeeb5d6ng7gxa477ylp4m

Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points [article]

Jiali Yan
2021 arXiv   pre-print
Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.  ...  In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on Sel^2(J) ×Sel^2(J) where J is the Jacobian variety of a genus two curve under the assumption  ...  Note that the local Cassels-Tate pairing depends on the choice of f ′ ∈ H 2 (G K , K(X) * ). We make the following remarks that are useful for the computation for the Cassels-Tate pairing.  ... 
arXiv:2109.08258v1 fatcat:ihue2ohtq5bgvandapbrgrnrhy

Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve [article]

Tom Fisher, Rachel Newton
2013 arXiv   pre-print
We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups.  ...  This requires significant modifications to both the local and global parts of the calculation.  ...  All computer calculations in Sections 3 and 4 were performed using Magma [5] . The second author is grateful for funding from DIAMANT.  ... 
arXiv:1306.1410v1 fatcat:3fuhg22xbfgkraz4tfs67tuura

Compact Hardware for Computing the Tate Pairing over 128-Bit-Security Supersingular Curves [chapter]

Nicolas Estibals
2010 Lecture Notes in Computer Science  
Hardware accelerators for pairing computation Pairings everywhere!  ...  work on FPGA implementations • low-security pairings  ...  Estibals -Compact hardware for 128-bit-security Tate pairing 14 / 28 Multiplication in the extension field Hardware Software F p F q = F p m·n F p m F q k Bottleneck of pairing computation  ... 
doi:10.1007/978-3-642-17455-1_25 fatcat:p2oq5yriurdvxovw5zilq3plsa

Hardware Acceleration of the Tate Pairing in Characteristic Three [chapter]

P. Grabher, D. Page
2005 Lecture Notes in Computer Science  
The core computational task is evaluation of a bilinear map, or pairing, over elliptic curves.  ...  Although identity based cryptography offers many functional advantages over conventional public key alternatives, the computational costs are significantly greater.  ...  Acknowledgements The authors would like to thank Rob Granger, Johann Großschädl, Elisabeth Oswald, Nigel Smart, Martijn Stam and Fré Vercauteren for invaluable help and support throughout the course of  ... 
doi:10.1007/11545262_29 fatcat:prtjbvglarh4nan47fpmi6x3d4

Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations [article]

Monique van Beek, Tom Fisher
2017 arXiv   pre-print
We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve.  ...  One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations.  ...  In practice we are happy to compute the Cassels-Tate pairing up to an overall scaling.  ... 
arXiv:1711.02432v1 fatcat:rgyxlrcefbhzndwci63mq3uo6e

Parallelizing the Weil and Tate Pairings [chapter]

Diego F. Aranha, Edward Knapp, Alfred Menezes, Francisco Rodríguez-Henríquez
2011 Lecture Notes in Computer Science  
This motivated us to consider the Weil pairing, whose potential speed advantages over the Tate pairing due to the absence of a final exponentiation in the former were first considered in [32] .  ...  is much faster, whereby the cost of managing the resulting threads dominates the cost of useful computation.  ...  Several variants of the Tate pairing have been defined in the literature, e.g., [6, 28, 34, 45, 39, 27] .  ... 
doi:10.1007/978-3-642-25516-8_17 fatcat:3eijkeemarebpesps6hn4ycue4
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