 Everyone, I'm Yu Wei. I'm from the state key laboratory of information security. In the Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China. It's my honor to introduce our paper on EuroCurved 2021. 2021. This paper is collaboration work with Professor Xu Guangwu. My topic today is pre-computation scheme of window-toe knife for acrobatic curves. The outline of my presentation is as follows. First, a brief introduction of this work. Secondly, we introduce from this map tau. Thirdly, we introduce the complex conjugate of tau and present efficient formulae of the complex conjugate. Fourthly, we propose a novel pre-computation scheme that we discuss scalar multiplication using the new proposed pre-computation scheme on cobalithic curves. Finally, we conclude this work. First, we introduce our work. The elliptical curve cryptography has drawn intensive attention from the literature. The family of cobalithic curves proposed by near-cobaliths are non-super-singular curves defined over binary fields. The arithmetic of cobalithic curves has been of theoretical and practical significance since the start of the elliptical curve cryptography. Four cobalithic curves were recommended to be used in digital signature standards. K is tabular on the schemes and K management. By NIST, this indicates that cobalithic curves can still be useful in practice. Cobalithic curves has a computational advantage that a fast scalar multiplication can be achieved by replacing point W with fobbing's map. The cobalithic curves are given as the following equation. These curves can be considered over the binary attention. Since EIF2 is a subgroup of EIF2 to the power of M, then there has a factor P. P is the cardinality of the main subgroup of the group of rational points on cobalithic curves. It's of cryptographic interest to choose suitable M that makes P a prime in the rest of our discussion. We just consider cases that P is a prime using the word conjecture in the range of 160 to 2000 when I is equal to 0. P is a prime when M is equal to 233 to 239. 277 to 283 and so on. When I is equal to 1, P is a prime when M is 163 to 283 and so on. For cobalithic curves with I equal to 0, I will be recommended by NIST. K233, K283, K2409 and K571. When N is represented as a binary representation, we usually employ Hohner's algorithm to calculate the scalar multiplication. An example of 31-time P is shown as follows. One can compute this scalar multiplication from left to right or right to left. Cobaliths propose a method of computing scalar multiplication using Froben's map. Salinas further developed an extremely efficient window-to-knife to compute scalar multiplication. There are many related works about the scalar multiplication. For cobalithic curves, Kher and Caesar discussed the multi-scalar multiplications. For cobalithic curves, on Eurocurb 2009, Chose Energy talked about the optimal pre-computation of window-to-knife for cobalithic curves. In 2016, in 2017, Kher proposed to achieve the mu4 norm form elliptical curves to speed up scalar multiplication or binary fields. To achieve the mu4 norm form elliptical curves can also be applied to cobalithic curves. This work will discuss the scalar multiplication using window-to-knife. We propose new formulae of complex conjugate of Froben's map. We design a novel pre-computation scheme and use it to improve the efficiency of scalar multiplication on cobalithic curves. The Froben's map talks in the morphism of EF2 to the power of M which is defined as the following formulae. For each point, we have this equation, that M is the main subgroup of the group of rational points of cobalithic curves, namely the subgroup of all the p. There are some properties of M. Also in this work, we mainly work on M. cobalithic curves are proposed a method of computing scalar multiplication, then time p, with p from the main subgroup of a cobalithic curve. And by representing it as a representation of Froben's map, Salinas further developed an extremely efficient window-to-knife to compute the scalar multiplication, refinements, and extensions of Salinas' methods were obtained by black, mochi, and sheen. The procedure of window-to-knife can be described as four steps. One, reduction. Two, window-to-knife with widest omib, pre-computation, and fourth, employ Hauner's algorithm to calculate scalar multiplication using window-to-knife and pre-computation. In 2017, Goheir introduced a twisted-mule phonon form elliptical curves over a binary field. Goheir proved that twisted-mule phonon form elliptical curves cover all the elliptical curves over binary fields recommended by NIST. A cobalithic curve using twisted-mule phonon form is called a mule phonon curve in this work. Because of its promising computational advantage, it's of great interest to consider the use of mule phonon curve in the window-to-knife, especially for the pre-computation part. Let tau bar be the complex conjugate of tau, and pb original points on a cobalithical curve, both events Dimitriev, dash, and psych. And dash, Goheir, and psych use complex multiplication in double-basic representation to speed up scalar multiplication and multi-scalar multiplication. Inspired by their elliptical results, we introduce new dreadings. Under these dreadings, we design new formulas which only require two multiplications and two squares. Chose lines to prove that one point addition is necessary for computing each pre-computation point. We use our new operation to replace point additions or mixed additions in pre-computation speed. As the cost of one addition costs more than our new operation, our formulas, there are quite a few field operations. Our formulas for our new operation, our new point operation, are part of w formula. It may lead to a simplicity of the implementation to take full advantage of our new operation. A new pre-computation scheme is developed to several more field operations. Our pre-computation scheme only requires six multiplications plus six field squares for window tall knives with widest fall. The cost of Salinas pre-computation scheme, the cost of Hengxing many days went down to pre-computation scheme. The cost of Chose the energy's pre-computation scheme and the cost of our pre-computation scheme are all in this table. The particle implementations show that our pre-computation is two times faster than Chose the energy's pre-computation. And at present, our pre-computation scheme is the state of art. In window tall knives, a bigger window wide, a bigger window wide corresponds to sparse tall expansion for scalar multiplication. However, when shooting to make the widest too big, as it would increase the pre-computation cost and affect the overall performance, currently, the state of art pre-computation scheme suggests to use widest at most six to achieve the best efficiency of scalar multiplication. Our pre-computation reduces the cost by half in most of the practical cases, namely, scheme with widest seven is about the same as the cost of existing pre-computation scheme. With widest six, this allows us to use a bigger window wide for example seven to get a faster scalar multiplication. The balance between the pre-computation part and the other part of scalar multiplication shows that the pre-computation takes a bigger radio of scalar multiplication than before. This is useful, especially for scalar multiplication with unfixed points. At present, scalar multiplication using our novel pre-computation scheme and with a bigger window wide is the state of art on Kabaly's curves. Now, we briefly conclude this work. This paper introduces new radius on Kabaly's curves. Efficient formulae of the new radius are then derived. And using a novel pre-computation scheme, our pre-computation scheme is about two times faster compared to the state of art technical of pre-computing scheme in the literature. The impact of our new efficient pre-computation is also reflected by the significant improvement of scalar multiplication traditionally. Window-toe knife with widest at most six is used to achieve the best scalar multiplication. Because of the dramatic cost reduction of the proposed pre-computation scheme, we are able to increase the widest for window-toe knife to seven for best scalar multiplication. This indicates that the pre-computation part becomes more important in performing scalar multiplication with our efficient pre-computation scheme and the new window-wides. Our scalar multiplication runs in at least 85% the time of Kabaly's work combining the best pre-computation our results push the scalar multiplication of Kabaly's books, a very well-studied and long-standing research area to a significant new stage. This is all I want to share. Any questions please email me. Thanks for your time. See you.