Paper 2019/1202

Rational isogenies from irrational endomorphisms

Wouter Castryck, Lorenz Panny, and Frederik Vercauteren


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.

Available format(s)
Public-key cryptography
Publication info
Published by the IACR in EUROCRYPT 2020
Isogeny-based cryptographyendomorphism ringsCSIDH
Contact author(s)
wouter castryck @ esat kuleuven be
lorenz @ yx7 cc
frederik vercauteren @ esat kuleuven be
2020-03-09: revised
2019-10-15: received
See all versions
Short URL
Creative Commons Attribution


      author = {Wouter Castryck and Lorenz Panny and Frederik Vercauteren},
      title = {Rational isogenies from irrational endomorphisms},
      howpublished = {Cryptology ePrint Archive, Paper 2019/1202},
      year = {2019},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.