Rational Isogenies from Irrational Endomorphisms

In this paper, we introduce a polynomial-time algorithm to compute a connecting minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathcal {O}$documentO-ideal between two supersingular elliptic curves over minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbb {F}_p$documentFp with common minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbb {F}_p$documentFp-endomorphism ring minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathcal {O}$documentO, 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.