This paper revisits the computation of pairings on a model of elliptic curve called Selmer curves. We extend the work of Zhang, Wang, Wang and Ye to the computation of other variants of the Tate pairing on this curve. Especially, we show that the Selmer model of an elliptic curve presents faster formulas for the computation of the Ate and optimal Ate pairings with respect to Weierstrass elliptic curves. We show how to parallelise the computation of these pairings and we obtained very fast results. We also present an example of optimal pairing on a pairing-friendly Selmer curve of embedding degree k = 12.
paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve . We improve
the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best result to date among curves with quartic
twists. For the doubling and addition steps in Miller's algorithm for the computation of the Tate pairing,
we obtain a theoretical gain up to and , depending on the embedding degree and the extension
field arithmetic, with respect to Weierstrass curves and previous results on Jacobi quartic curves. Furthermore and for the first time, we compute and implement Ate, twisted Ate and optimal pairings on the Jacobi quartic curves. Our results are up to more efficient compared to the case of Weierstrass curves with
Barreto P. S. L. M. Naehrig M. Pairing-friendly elliptic curves of prime order Selected Areas in Cryptography SAC 2005 Lecture Notes in Comput. Sci. 3897 Springer Berlin 2006 319 331 4 J. Beuchat, J. E. González-Díaz, S. Mitsunari, E. Okamoto, F. Rodríguez-Henríquez and T. Teruya,
High-speed software implementation of the optimal Atepairing over Barreto–Naehrig curves,
Pairing-Based Cryptography (Pairing 2010),
Lecture Notes in Comput. Sci. 6487,
Springer, Berlin (2010), 21–39.
Beuchat J. González-Díaz J. E. Mitsunari S. Okamoto E. Rodríguez
both in RNS and in radix representation. So we did
not use it in this work.
72 S. Duquesne
4.5 Other pairings
Recently, some variants of the Tate pairing appear in the literature. The main goal
is to reduce the length of the Miller loop. The price to be paid is that the points P
and Q are swapped. This means that the elliptic curve arithmetic will hold in Fpk
(or Fpk=d for twisted versions) instead of Fp so that even if the number of steps in
the Miller loop will decrease, the cost of each step will increase. We give here the
Ate and R-Atepairings.
previous work of
Zhao, Zhang and Xie .
There has been a lot of work on efficient implementations of the general bilin-
ear pairing e.P;Q/. Motivated by the idea of Miller loop shortening , many
optimizations have been proposed [6, 7, 9, 14]. On the other hand, pairings on
hyperelliptic curves have been also investigated. Some excellent surveys can be
found in [1,3]. It should be remarked that the Eta and Atepairings on hyperelliptic
curves have been also presented in  and , respectively. However, there is little
work on the performance of self-pairings [12
A class of hash functions based on the algebraic
Gennady A. Noskov, Alexander N. Rybalov
Generic case complexity of the Graph Isomorphism
Murray Elder, Jennifer Taback
Thompson’s group F is 1-counter graph automatic 21
The automorphism group of a finitely generated virtually
abelian group 35
Omar Akchiche, Omar Khadir
Factoring multi-power RSA moduli with primes sharing
least or most significant bits 47
Emmanuel Fouotsa, Abdoul Aziz Ciss
Faster Atepairing computation on Selmer’s model of
elliptic curves 55
Kenneth R. Blaney
md−1 = zq
md+1−2 = z−2.
Since gcd(qmd + 1, qmd − 1) | 2 and equals 2 if and only if q is odd, it follows
that one can obtain a well-defined bilinear pairing without a final exponentiation if q is
even. If q is odd then a squaring is required.
Granger et al  give a more general derivation of these results in the case of the
atepairing. With their approach the final exponentiation is never required, even if q is
3.3 The characteristic two case
We illustrate the above ideas in the case of supersingular elliptic curves in characteristic
2. The curve E
account for the in-
fluence of possible homogeneous material layers above and
below the FE unit cell.
According to the above described solution procedure, the
FE unit cell is to excite correspondingly which can be in
form of impressed volume currents or impressed electrical
field strengths. This is realisable by discrete ports in a full-
wave driven simulation. The electric field strength as well
as any other kind of integral measure can be observed at
individual locations. To accelerate the search of appropri-
atepairs of .!;kt00/, i.e. the eigenvalue, bisectional
05 and 0 6 of 8 exhibit no pairing properties, cf.
Fig. 5. C onsequently 0 h-0 , and hence Khj of 1 are
small. Thus Shi of 1 is considerably sm aller than that
of polyacenes.This is explicitely taken into account in
our regression lines for 1—6. O bviously two features
are responsible for the deep blue colour of 1 : the
absence of at least an approxim atepairing rela tion
ship betw een 0 h and 0 , in 1 which is also responsible
for the blue colour of azulene , and the occurence
of heteroatom s in such a m anner tha t the small E 56
value of 8 is re
atepairs increases. The more variables are introduced the higher the dissimilarity,
and the4cor eventually overlaps the permutation results.
In contrast with the results in Figure 1(b), the variation in the 4cor was much
bigger; this variation was also seen in simulation studies where the measurement
error was increased (Figures B1-B4 (Appendix B)).
The pair of canonical variates with the smallest4cor (0.338) associated 10 gene
copy number variables with 20 gene expression variables, with an average canon-
ical correlation of 0.970. The pair of canonical