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Paper: Rational isogenies from irrational endomorphisms

Authors:
Wouter Castryck , Cosic, research group at imec and KU Leuven, Belgium
Lorenz Panny , Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, The Netherlands
Frederik Vercauteren , Cosic, research group at imec and KU Leuven, Belgium
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DOI: 10.1007/978-3-030-45724-2_18 (login may be required)
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Conference: EUROCRYPT 2020
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.
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BibTeX
@inproceedings{eurocrypt-2020-30243,
  title={Rational isogenies from irrational endomorphisms},
  booktitle={39th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zagreb, Croatia, May 10–14, 2020, Proceedings},
  series={Lecture Notes in Computer Science},
  publisher={Springer},
  keywords={isogeny-based cryptography;endomorphism rings;CSIDH},
  volume={12105},
  doi={10.1007/978-3-030-45724-2_18},
  author={Wouter Castryck and Lorenz Panny and Frederik Vercauteren},
  year=2020
}