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Abstract

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.

Abstract

This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y2=dX4+Z4. 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 27% and 39%, 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 27% more efficient compared to the case of Weierstrass curves with quartic twists.

(2006), 319–331. 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 Ate pairing 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-Ate pairings. The Ate pairing

previous work of Zhao, Zhang and Xie [17]. 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 [2], 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 Ate pairings on hyperelliptic curves have been also presented in [2] and [5], respectively. However, there is little work on the performance of self-pairings [12

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. Gunnells A class of hash functions based on the algebraic eraser™ 1 Gennady A. Noskov, Alexander N. Rybalov Generic case complexity of the Graph Isomorphism Problem 9 Murray Elder, Jennifer Taback Thompson’s group F is 1-counter graph automatic 21 Bettina Eick 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 Ate pairing computation on Selmer’s model of elliptic curves 55 Kenneth R. Blaney

so zq 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 [11] give a more general derivation of these results in the case of the ate pairing. With their approach the final exponentiation is never required, even if q is odd. 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- ate pairs 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 ate pairing rela tion­ ship betw een 0 h and 0 , in 1 which is also responsible for the blue colour of azulene [15], and the occurence of heteroatom s in such a m anner tha t the small E 56 value of 8 is re

canonical vari- ate pairs 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