Failing to Hash Into Supersingular Isogeny Graphs

Jeremie Booher, Ross Bowden, Javad Doliskani, Tako Boris Fouotsa, Steven D. Galbraith, Sabrina Kunzweiler, Simon-Philipp Merz, Christophe Petit, Benjamin Smith, Katherine E. Stange*, Yan Bo Ti, Christelle Vincent, José Felipe Voloch, Charlotte Weitkämper, Lukas Zobernig

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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Abstract

An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of ‘hard supersingular curves’ that is equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular -isogeny graph, which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd’s of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces, and applying Kummer surfaces and (v) using quantum random walks.

Original languageEnglish
Pages (from-to)2702-2719
Number of pages18
JournalThe Computer Journal
Volume67
Issue number8
Early online date24 May 2024
DOIs
Publication statusPublished - Aug 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024. Published by Oxford University Press on behalf of The British Computer Society. All rights reserved.

Keywords

  • elliptic curves
  • hashing
  • isogeny-based cryptography

ASJC Scopus subject areas

  • General Computer Science

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