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

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.

To appear in Advances in Mathematics of Communications