The Generalized Montgomery Coordinate: A New Computational Tool for Isogeny-based Cryptography

The Generalized Montgomery Coordinate: A New Computational Tool for Isogeny-based Cryptography
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发表时间：
2022
期刊：
IACR Cryptol. ePrint Arch.
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通讯作者：
Tomoki Moriya;Hiroshi Onuki;Yusuke Aikawa;T. Takagi
Tomoki Moriya;Hiroshi Onuki;Yusuke Aikawa;T. Takagi
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其他
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作者：
Tomoki Moriya;Hiroshi Onuki;Yusuke Aikawa;T. Takagi

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.近年来，一些研究构造了椭圆曲线上的单坐标算法。例如，给出了蒙哥马利曲线的x坐标、蒙哥马利−曲线的x坐标、Edwards曲线的w坐标、Hu（cid：11）曲线的w坐标、扭曲Jacobi交点的ω坐标等公式。这些公式是有用的同源密码学，因为它们的紧凑性和e（cid：14）的科学。在本文中，我们在椭圆曲线上定义了一个新的函数，称为广义蒙哥马利坐标，它具有上述的（cid：12）ve坐标作为特例。对于广义蒙哥马利坐标，我们构造了一个标量乘的显式公式，其中包括除多项式，以及等距点下的像点公式和余域曲线的系数公式.最后，我们提出了两个应用的理论的广义蒙特-布尔坐标。第一个是构造了一个新的计算蒙哥马利曲线的等距线的有效公式。在我们的实现中，这个公式比以前的一个公式更有效。第二个是构造一个新的用于CSURF的蒙哥马利-曲线的广义蒙哥马利坐标。
. Recently, some studies have constructed one-coordinate arith-metics on elliptic curves. For example, formulas of the x -coordinate of Montgomery curves, x -coordinate of Montgomery − curves, w -coordinate of Edwards curves, w -coordinate of Hu(cid:11)’s curves, ω -coordinates of twisted Jacobi intersections have been proposed. These formulas are useful for isogeny-based cryptography because of their compactness and e(cid:14)ciency. In this paper, we de(cid:12)ne a novel function on elliptic curves called the generalized Montgomery coordinate that has the (cid:12)ve coordinates described above as special cases. For a generalized Montgomery coordinate, we construct an explicit formula of scalar multiplication that includes the division polynomial, and both a formula of an image point under an isogeny and that of a coe(cid:14)cient of the codomain curve. Finally, we present two applications of the theory of a generalized Mont-gomery coordinate. The (cid:12)rst one is the construction of a new e(cid:14)cient formula to compute isogenies on Montgomery curves. This formula is more e(cid:14)cient than the previous one for high degree isogenies as the √ (cid:19)elu’s formula in our implementation. The second one is the construction of a new generalized Montgomery coordinate for Montgomery − curves used for CSURF.