论文标题
较大的质量质量基因的更快计算
Faster computation of isogenies of large prime degree
论文作者
论文摘要
令$ \ mathcal {e}/\ mathbb {f} _q $为椭圆曲线，而$ \ mathcal {e}中的$ p $ a点（\ Mathbb {f} _Q）prime prime $ \ ell $ \ ell $。 Vélu的公式让我们计算一条商曲线$ \ MATHCAL {E}'= \ MATHCAL {e}/\ langle {p} \ rangle $和有理图，定义了标准的nisogeny $ $ ϕ：\ MATHCAL {E} $ \ mathbb {f} _q $ - operations，其中$ \ tilde {o} $在$ q $中是均匀的。本文显示了如何计算$ \ nathcal {e} $，而$ d（q）$ for $ q $ for $ q $ in $ \ nathcal in $ \ nathcal {e}（\ nathbbbbbbbbbbbbbbbbbbbbbb c） $ \ tilde {o}（\ sqrt {\ ell}）$ $ \ mathbb {f} _q $ - operations，其中$ \ tilde {o} $在$ q $中再次均匀。
Let $\mathcal{E}/\mathbb{F}_q$ be an elliptic curve, and $P$ a point in $\mathcal{E}(\mathbb{F}_q)$ of prime order $\ell$. Vélu's formulae let us compute a quotient curve $\mathcal{E}' = \mathcal{E}/\langle{P}\rangle$ and rational maps defining a quotient isogeny $ϕ: \mathcal{E} \to \mathcal{E}'$ in $\tilde{O}(\ell)$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is uniform in $q$.This article shows how to compute $\mathcal{E}'$, and $ϕ(Q)$ for $Q$ in $\mathcal{E}(\mathbb{F}_q)$, using only $\tilde{O}(\sqrt{\ell})$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is again uniform in $q$.As an application, this article speeds up some computations used in the isogeny-based cryptosystems CSIDH and CSURF.
