This is the discussion thread for the fourth session of season four of The Isogeny Club, where Arthur presents his talk titled: **The endomorphism ring problem given one endomorphism**

**Abstract**

Computing the endomorphism ring of elliptic curves is a fundamental problem in isogeny-based cryptography as the security of most schemes in this field relies on its hardness. How much easier does this problem get when a public endomorphism is provided? In this talk, we answer this question by giving a rigorous complexity analysis of the endomorphism ring problem given one endomorphism. To this end, we review how higher-dimensional isogenies can be used to divide endomorphisms by integers in polynomial time. Along the way, we also present improvements on previous results regarding oriented variants of the endomorphism ring problem.

**Video**