The supersingular isogeny path and endomorphism ring problems are equivalent

Benjamin Wesolowski

Paper 2021/919

The supersingular isogeny path and endomorphism ring problems are equivalent

Benjamin Wesolowski

Abstract

We prove that the path-finding problem in $ℓ$ -isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis. The presumed hardness of these problems is foundational for isogeny-based cryptography. As an essential tool, we develop a rigorous algorithm for the quaternion analog of the path-finding problem, building upon the heuristic method of Kohel, Lauter, Petit and Tignol. This problem, and its (previously heuristic) resolution, are both a powerful cryptanalytic tool and a building-block for cryptosystems.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published elsewhere. Minor revision. FOCS 2021
Keywords: isogeny-based cryptography cryptanalysis endomorphism ring isogeny path
Contact author(s): benjamin wesolowski @ math u-bordeaux fr
History: 2021-09-10: revised; 2021-07-09: received; See all versions
Short URL: https://ia.cr/2021/919
License: CC BY

BibTeX

@misc{cryptoeprint:2021/919,
      author = {Benjamin Wesolowski},
      title = {The supersingular isogeny path and endomorphism ring problems are equivalent},
      howpublished = {Cryptology {ePrint} Archive, Paper 2021/919},
      year = {2021},
      url = {https://eprint.iacr.org/2021/919}
}