**Rational isogenies from irrational endomorphisms**

*Wouter Castryck and Lorenz Panny and Frederik Vercauteren*

**Abstract: **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.

**Category / Keywords: **public-key cryptography / Isogeny-based cryptography, endomorphism rings, CSIDH

**Date: **received 14 Oct 2019

**Contact author: **wouter castryck at esat kuleuven be,lorenz@yx7 cc,frederik vercauteren@esat kuleuven be

**Available format(s): **PDF | BibTeX Citation

**Version: **20191015:075059 (All versions of this report)

**Short URL: **ia.cr/2019/1202

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