An Effective Lower Bound on the Number of Orientable Supersingular Elliptic Curves

In this article, we prove a generic lower bound on the number of \(\mathfrak {O}\) -orientable supersingular curves over \(\mathbb {F}_{p^2}\) , i.e. curves that admit an embedding of the quadratic order \(\mathfrak {O}\) inside their endomorphism ring. Prior to this work, the only known effective lower-bound is restricted to small discriminants. Our main result targets the case of fundamental discriminants and we derive a generic bound using the expansion properties of the supersingular isogeny graphs. Our work is motivated by isogeny-based cryptography and the increasing number of protocols based on \(\mathfrak {O}\) -oriented curves. In particular, our lower bound provides a complexity estimate for the brute-force attack against the new \(\mathfrak {O}\) -uber isogeny problem introduced by De Feo, Delpech de Saint Guilhem, Fouotsa, Kutas, Leroux, Petit, Silva and Wesolowski in their recent article on the SETA encryption scheme.