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arXiv · 2203.02097

The Endomorphism Rings of Supersingular Elliptic Curves over $\mathbb{F}_p$ and the Binary Quadratic Forms

Guanju Xiao·Zijian Zhou·Yingpu Deng·Longjiang Qu
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Abstract

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.

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Guanju Xiao, Zijian Zhou, Yingpu Deng, Longjiang Qu. 2022-03-04. The Endomorphism Rings of Supersingular Elliptic Curves over $\mathbb{F}_p$ and the Binary Quadratic Forms. https://arxiv.org/abs/2203.02097

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