On collisions related to an ideal class of order 3 in CSIDH (Q1995506)

The authors of this paper study an isogeny-based key exchange for post quantum cryptography. It is based on the action of an ideal class group on isomorphism classes of supersingular elliptic curves. In the key exchange, the classes are represented by vectors with integer coefficients and the number of ideal classes is related to the security level ofthe system. They study the correspondence between the integer vectors and the ideal classes and proof that the all ones vector corresponds to an ideal class of order 3 and, therefore, distinct vectors belongs to the same ideal class. This means that distinct secret keys correspond to the same public key. In the final part of the paper, they show a new ideal representation that does not include those collisions. For the entire collection see [Zbl 1454.68007].