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research article

Failing to Hash Into Supersingular Isogeny Graphs

Booher, Jeremy
•
Bowden, Ross
•
Doliskani, Javad
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May 24, 2024
Computer Journal

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 $\ell $ -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.

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Type
research article
DOI
10.1093/comjnl/bxae038
Web of Science ID

WOS:001230642700001

Author(s)
Booher, Jeremy
Bowden, Ross
Doliskani, Javad
Boris Fouotsa, Tako  
Galbraith, Steven D.
Kunzweiler, Sabrina
Merz, Simon-Philipp
Petit, Christophe
Smith, Benjamin
Stange, Katherine E.
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Date Issued

2024-05-24

Publisher

Oxford Univ Press

Published in
Computer Journal
Subjects

Technology

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Isogeny-Based Cryptography

•

Hashing

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Elliptic Curves

Editorial or Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
LASEC  
FunderGrant Number

Marsden Fund Council

21w5229

Available on Infoscience
June 19, 2024
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/208624
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