Rational isogenies from irrational endomorphisms.

In this paper, we introduce a polynomial-time algorithm to compute a connecting \(\mathcal {O}\)-ideal between two supersingular elliptic curves over \(\mathbb {F}_p\) with common \(\mathbb {F}_p\)-endomorphism ring \(\mathcal {O}\), given a description of their full endomorphism rings. This algorithm provides a reduction of the security of the CSIDH cryptosystem to the problem of computing endomorphism rings of supersingular elliptic curves. A similar reduction for SIDH appeared at Asiacrypt 2016, but relies on totally different techniques. Furthermore, we also show that any supersingular elliptic curve constructed using the complex-multiplication method can be located precisely in the supersingular isogeny graph by explicitly deriving a path to a known base curve. This result prohibits the use of such curves as a building block for a hash function into the supersingular isogeny graph.