求大度同构的次指数算法
A Subexponential Algorithm for Evaluating Large Degree Isogenies
摘要：
椭圆曲线之间的等距是一个代数态射，它是一个群同态。密码学中的许多应用都要求有效地计算椭圆曲线之间的大度数同构。对于同一自同态环的普通曲线，以前最好的已知算法具有最坏情况下的运行时间，其与输入长度成指数关系。在本文中，我们证明了这个问题可以解决在亚指数时间下合理的算法。我们的方法是基于因式分解的理想对应的内核的isogenies，模主理想，到一个产品的较小的素理想的isogenies可以直接计算。结合Bostan等人以前的工作，我们的算法在仅给定起始曲线和核的准最优时间内产生大度同构的方程。
英文摘要：
An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous best known algorithm has a worst case running time which is exponential in the length of the input. In this paper we show this problem can be solved in subexponential time under reasonable heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.
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