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Faster computation of the Tate pairing

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*. ...

##
###
Faster computation of the Tate pairing
[article]

2010
*
arXiv
*
pre-print

This paper proposes new explicit formulas for

arXiv:0904.0854v3
fatcat:utbwvxy4inghphsllhnwn2gnhi
*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*. ...##
###
TinyTate: Computing the Tate Pairing in Resource-Constrained Sensor Nodes

2007
*
Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007)
*

On

doi:10.1109/nca.2007.48
dblp:conf/nca/OliveiraAMDLD07
fatcat:gs6anqesdzhireq5hgh32chrja
*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. ...##
###
Refinements of Miller's algorithm for computing the Weil/Tate pairing

2006
*
Journal of Algorithms
*

In this paper we propose three refinements to Miller's algorithm for

doi:10.1016/j.jalgor.2005.01.009
fatcat:sahhckccpvc5rpgzkaii7ddb3a
*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 . ...##
###
Efficient Computations of the Tate Pairing for the Large MOV Degrees
[chapter]

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. ...

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

2021
*
arXiv
*
pre-print

We also include a worked example demonstrating we can turn

arXiv:2109.08257v1
fatcat:urdp3r7vhrdf7kn6qaxht25cgy
*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). ...##
###
The yoga of the Cassels–Tate pairing

2010
*
LMS Journal of Computation and Mathematics
*

AbstractCassels has described a

doi:10.1112/s1461157010000185
fatcat:3jofamomvzb4tgk7n4tqlmohkm
*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. ...##
###
Implementing the Tate Pairing
[chapter]

2002
*
Lecture Notes in Computer Science
*

To efficiently implement these cryptosystems it is necessary to optimise

doi:10.1007/3-540-45455-1_26
fatcat:6ny5mblrdjcdfgbymgtlcf56my
*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. ...##
###
TATE PAIRING COMPUTATION ON THE DIVISORS OF HYPERELLIPTIC CURVES OF GENUS 2

2008
*
Journal of the Korean Mathematical Society
*

Furthermore, we obtain a very efficient algorithm for

doi:10.4134/jkms.2008.45.4.1057
fatcat:uennwyeeb5d6ng7gxa477ylp4m
*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. ...##
###
Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points
[article]

2021
*
arXiv
*
pre-print

Finally, we include a worked example demonstrating we can indeed improve

arXiv:2109.08258v1
fatcat:ihue2ohtq5bgvandapbrgrnrhy
*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*. ...##
###
Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve
[article]

2013
*
arXiv
*
pre-print

We extend

arXiv:1306.1410v1
fatcat:3fuhg22xbfgkraz4tfs67tuura
*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. ...##
###
Compact Hardware for Computing the Tate Pairing over 128-Bit-Security Supersingular Curves
[chapter]

2010
*
Lecture Notes in Computer Science
*

Hardware accelerators for

doi:10.1007/978-3-642-17455-1_25
fatcat:p2oq5yriurdvxovw5zilq3plsa
*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*...##
###
Hardware Acceleration of the Tate Pairing in Characteristic Three
[chapter]

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 ...

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

2017
*
arXiv
*
pre-print

We explain a method for

arXiv:1711.02432v1
fatcat:rgyxlrcefbhzndwci63mq3uo6e
*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. ...##
###
Parallelizing the Weil and Tate Pairings
[chapter]

2011
*
Lecture Notes in Computer Science
*

This motivated us to consider

doi:10.1007/978-3-642-25516-8_17
fatcat:3eijkeemarebpesps6hn4ycue4
*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] . ...
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