Endomorphism rings of supersingular elliptic curves over Fₚ and binary quadratic forms

It is well known that there is a one-to-one correspondence between supersingular j-invariants up to the action of Gal(Fp²/Fₚ) and type classes of maximal orders in Bp, ∞ by Deuring's theorem. Interestingly, we establish a one-to-one correspondence between Fₚ-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 Fₚ-isogenies between elliptic curves could be represented by quadratic forms, we show that actions of these isogenies on supersingular elliptic curves over Fₚ are compatible with the composition of quadratic forms. Based on these results, we reduce the security of CSIDH cryptosystem to computing this correspondence explicitly.