素数阶椭圆曲线的覆盖攻击
Cover attacks for elliptic curves with prime order
AI总结：
提出一种经 Weil 限制、$(\ell,\ell,\ell)$-同源迁移至三亏格 Jacobian 并用指标演算的新型覆盖攻击，可在 $\tilde{O}(q)$ 时间内求解部分素数阶椭圆曲线离散对数。
摘要：
我们给出了一种求解三次扩域 $\mathbb{F}_{q^3}$ 上椭圆曲线离散对数问题的新方法。该方法基于一种迁移：首先，从所考虑椭圆曲线关于 $\mathbb{F}_{q^3}/\mathbb{F}_q$ 的 Weil 限制，到 $\mathbb{F}_q$ 上一条三亏格曲线的 Jacobian 簇，施加一个 $\mathbb{F}_q$-有理 $(\ell,\ell,\ell)$-同源；随后在该 Jacobian 中通过指标演算攻击求解问题。尽管在所期望同态的构造中没有使用覆盖映射，但从某种意义上说，该方法是一种覆盖攻击。因此，可以在 $\tilde{O}(q)$ 时间内求解 $\mathbb{F}_{q^3}$ 上某些素数阶椭圆曲线群中的离散对数问题。
英文摘要：
We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields $\mathbb{F}_{q^3}$. It is based on a transfer: First an $\mathbb{F}_q$-rational $(\ell,\ell,\ell)$-isogeny from the Weil restriction of the elliptic curve under consideration with respect to $\mathbb{F}_{q^3}/\mathbb{F}_q$ to the Jacobian variety of a genus three curve over $\mathbb{F}_q$ is applied and then the problem is solved in the Jacobian via the index-calculus attacks. Although using no covering maps in the construction of the desired homomorphism, this method is, in a sense, a kind of cover attack. As a result, it is possible to solve the discrete logarithm problem in some elliptic curve groups of prime order over $\mathbb{F}_{q^3}$ in a time of $\tilde{O}(q)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
