3-超奇异Edwards曲线的5-等势线
3- and 5-Isogenies of Supersingular Edwards Curves
摘要：
本文分析了完全和二次超奇异Edwards曲线的3-和5-同构的性质和存在条件。对于基于SIDH算法的密钥封装，提出了使用最小奇数度3和5的同构，这允许绕过2阶和4阶奇异点的问题，2-同构的特征。综述了简单域上的完全、二次和扭曲Edwards曲线类的主要性质。奇数度的等距方程被简化为适合于维尔斯特拉斯形式的曲线的形式。要做到这一点，在广义爱德华兹形式中使用曲线点的修正加法定律，它保持了曲线返回点的水平对称性。本文给出了小简单域上完全Edwards超奇异曲线的3-和5-同构的计算实例，并讨论了它们在高阶核下的同构复合的性质。给出了在射影坐标系下完全Edwards曲线类和二次Edwards曲线类中计算奇次3和奇次5的等距线的上复杂度估计公式，并构造了计算Edwards曲线类的3-和5-等距线的算法，其复杂度分别为6 M + 4S和12 M + 5S.给出了4 × 3 m × 5 n阶和8 × 3 m × 5 n阶超奇异完全Edwards曲线和二次Edwards曲线存在的条件，并确定了在128比特量子安全水平上实现SIDH算法时密码体制的一些参数.
英文摘要：
An analysis is made of the properties and conditions for the existence of 3- and 5-isogenies of complete and quadratic supersingular Edwards curves. For the encapsulation of keys based on the SIDH algorithm, it is proposed to use isogeny of minimal odd degrees 3 and 5, which allows bypassing the problem of singular points of the 2nd and 4th orders, characteristic of 2-isogenies. A review of the main properties of the classes of complete, quadratic, and twisted Edwards curves over a simple field is given. Equations for the isogeny of odd degrees are reduced to a form adapted to curves in the form of Weierstrass. To do this, use the modified law of addition of curve points in the generalized Edwards form, which preserves the horizontal symmetry of the curve return points. Examples of the calculation of 3- and 5-isogenies of complete Edwards supersingular curves over small simple fields are given, and the properties of the isogeny composition for their calculation with large-order kernels are discussed. Equations are obtained for upper complexity estimates for computing isogeny of odd degrees 3 and 5 in the classes of complete and quadratic Edwards curves in projective coordinates; algorithms are constructed for calculating 3- and 5-isogenies of Edwards curves with complexity 6M + 4S and 12M + 5S, respectively. The conditions for the existence of supersingular complete and quadratic Edwards curves of order 4x3mx5n and 8x3mx5n are found. Some parameters of the cryptosystem are determined when implementing the SIDH algorithm at the level of quantum security of 128 bits.
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