The Endomorphism Rings of Supersingular Elliptic Curves over \mathbbF_p and the Binary Quadratic Forms

2022/03/04 by Xiao, Guanju, Zhou, Zijian, Deng, Yingpu +1
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2203.02097

Abstract

It is well known that there is a one-to-one correspondence between supersingular j-invariants up to the action of Gal(\mathbbF_p²/\mathbbF_p) and type classes of maximal orders in B_p,∞ by Deuring's theorem. Interestingly, we establish a one-to-one correspondence between \mathbbF_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 \mathbbF_p-isogenies between elliptic curves could be represented by quadratic forms, we show that operations of these isogenies on supersingular elliptic curves over \mathbbF_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.

Related

• Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms
• Gross lattices of supersingular elliptic curves
• Oriented Supersingular Elliptic Curves and Eichler Orders
• On the Correspondence between Supersingular Elliptic Curves and maximal quaternionic Orders
• Endomorphism rings of supersingular elliptic curves over \mathbbF_p