The security of public-key systems is based on the difficulty of solving certain mathematical problems. With the possible emergence of large-scale quantum computers several of these problems, such as factoring and computing discrete logarithms, would be efficiently solved. Research on quantum-resistant public-key cryptography, also called post-quantum cryptography (PQC), has been productive in recent years. Public-key cryptosystems based on the problem of computing isogenies between supersingular elliptic curves appear to be good candidates for the next generation of public-key cryptography standards in the PQC scenario. In this work, motivated by a previous work by D. Moody and D. Shumow [17], we derived maps for elliptic curves represented in Jacobi Intersection and Twisted Hessian models. Our derivation follows a multiplicative strategy that contrasts with the additive idea presented in the Vélu formula. Finally, we present a comparison of computational cost to generate maps for isogenies of degree $ l $, where $ l = 2k + 1 $. In affine coordinates, our formulas require $ 46.8 \% $ less computation than the Huff model and $ 48\% $ less computation than the formulas given for the Extended Jacobi Quartic model when computing isogenies of degree $ 3 $. Considering higher degree isogenies as $ 101 $, our formulas require $ 23.4\% $ less computation than the Huff model and $ 24.7 \% $ less computation than the formula for the Extended Jacobi Quartic model.

中文翻译：

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Jacobi交点和扭曲的粗麻线的等距公式

公钥系统的安全性基于解决某些数学问题的难度。随着大规模量子计算机的出现，这些问题中的几个问题，例如因数分解和离散对数的计算，将得到有效解决。近年来，对量子抗性公钥密码学（也称为后量子密码学（PQC））的研究取得了丰硕的成果。基于超奇异椭圆曲线之间的计算同质性问题的公共密钥密码系统似乎是PQC场景中下一代公共密钥密码标准的理想候选者。在这项工作中，受到D. Moody和D. Shumow [17]，我们得出了以Jacobi交集和Twisted Hessian模型表示的椭圆曲线的映射。我们的推导遵循乘性策略，该策略与Vélu公式中提出的加性概念形成对比。最后，我们对计算成本进行比较，以生成度为$ l $的等价图的映射，其中$ l = 2k + 1 $。在仿射坐标中，当计算等价度$ 3 $时，我们的公式所需的计算量比Huff模型少$ 46.8 \％$，并且比扩展的Jacobi Quartic模型给出的公式少$ 48 \％$。考虑到较高的同质度为$ 101 $，我们的公式比Huff模型所需的计算少$ 23.4 \％$，而与Extended Jacobi Quartic模型的公式相比，则需要少$ 24.7 \％$的计算。