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It is well known that there is a one-to-one correspondence betweensupersingular $j$-invariants up to the action of$\text{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)$ and type classes of maximal ordersin $B_{p,\infty}$ by Deuring's theorem. Interestingly, we establish aone-to-one correspondence between $\mathbb{F}_p$-isomorphism classes ofsupersingular elliptic curves and primitive reduced binary quadratic forms withdiscriminant $-p$ or $-16p$. Due to this correspondence and the fact that$\mathbb{F}_p$-isogenies between elliptic curves could be represented byquadratic forms, we show that operations of these isogenies on supersingularelliptic curves over $\mathbb{F}_p$ are compatible with the composition ofquadratic forms. Based on these results, we could reduce the security of CSIDHcryptosystem to computing this correspondence explicitly.

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