How to Construct CSIDH on Edwards Curves
How to Construct CSIDH on Edwards Curves
复制标题
如何在爱德华兹曲线上构造CSIDH
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
T. Takagi
中科院分区：
文献类型：
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作者：
Tomoki Moriya;Hiroshi Onuki;T. Takagi
CSIDH是Castryck、Lange、Martindale、Panny和Renes于2018年提出的基于等基因的密钥交换协议。CSIDH是基于Montgomery曲线的(mathbb {F}_p)-同构类上的理想类群作用。为了计算类组动作，我们需要取（mathbb {F}_{p^2}）上定义的点。原始的CSIDH算法需要通过将点表示为蒙哥马利曲线上的x坐标来计算（mathbb {F}_p）。Meyer和Reith在2018年提出了一种更快的CSIDH算法，该算法通过使用蒙哥马利曲线和爱德华兹曲线之间的双族图来计算爱德华兹曲线上的等基因。在Edwards曲线上有一个特殊的坐标（w坐标）来计算群运算和等同性。如果我们试图用类似于Montgomery曲线的w坐标来计算Edwards曲线上的类群作用，我们必须考虑在（mathbb {F}_{p^4}）上定义的点。因此，仅在（mathbb {F}_p）上计算w坐标的Edwards曲线上的类群作用并不是一项简单的任务。
CSIDH is an isogeny-based key exchange protocol proposed by Castryck, Lange, Martindale, Panny, and Renes in 2018. CSIDH is based on the ideal class group action on (mathbb {F}_p)-isomorphism classes of Montgomery curves. In order to calculate the class group action, we need to take points defined over (mathbb {F}_{p^2}). The original CSIDH algorithm requires a calculation over (mathbb {F}_p) by representing points as x-coordinate over Montgomery curves. Meyer and Reith proposed a faster CSIDH algorithm in 2018 which calculates isogenies on Edwards curves by using a birational map between a Montgomery curve and an Edwards curve. There is a special coordinate on Edwards curves (the w-coordinate) to calculate group operations and isogenies. If we try to calculate the class group action on Edwards curves by using the w-coordinate in a similar way on Montgomery curves, we have to consider points defined over (mathbb {F}_{p^4}). Therefore, it is not a trivial task to calculate the class group action on Edwards curves with w-coordinates over only (mathbb {F}_p).
DOI：
10.2140/obs.2020.4.39
发表时间：
2020-03
期刊：
ArXiv
影响因子：
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作者：
D. Bernstein;L. D. Feo;Antonin Leroux;Benjamin A. Smith
通讯作者：
D. Bernstein;L. D. Feo;Antonin Leroux;Benjamin A. Smith