论文标题
$ \ mathbb {f} _p $和二进制二进制形式的上椭圆形曲线的内态环
The Endomorphism Rings of Supersingular Elliptic Curves over $\mathbb{F}_p$ and the Binary Quadratic Forms
论文作者
论文摘要
众所周知，在$ \ text {gal}（\ Mathbb {f} _ {p^2}/\ mathbb {f} _p）_p {p^2}/\ mathbb {f} _p）$和最大级别类型的$ b _ {p p p p，p，p，f，f，f，f，f，ift ifty}中。有趣的是，我们在$ \ mathbb {f} _p $ -isomorphism类之间建立了一对一的对应关系，用于超级椭圆曲线的类别和带有判别$ -P $或$ -16p $的原始减少二进制二进制形式。由于这种对应关系以及椭圆形曲线之间的$ \ mathbb {f} _p $ - 异生，可以用二次形式表示，我们表明，这些同等基因在$ \ mathbb {f} _p $上与Quadrats形式的组成兼容。基于这些结果，我们可以将CSIDH密码系统的安全性降低到明确计算此通信的安全性。
It is well known that there is a one-to-one correspondence between supersingular $j$-invariants up to the action of $\text{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)$ and type classes of maximal orders in $B_{p,\infty}$ by Deuring's theorem. Interestingly, we establish a one-to-one correspondence between $\mathbb{F}_p$-isomorphism classes of supersingular elliptic curves and primitive reduced binary quadratic forms with discriminant $-p$ or $-16p$. Due to this correspondence and the fact that $\mathbb{F}_p$-isogenies between elliptic curves could be represented by quadratic forms, we show that operations of these isogenies on supersingular elliptic curves over $\mathbb{F}_p$ are compatible with the composition of quadratic forms. Based on these results, we could reduce the security of CSIDH cryptosystem to computing this correspondence explicitly.
