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Failing to Hash Into Supersingular Isogeny Graphs.

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Title: Failing to Hash Into Supersingular Isogeny Graphs.
Authors: Booher, Jeremy1, Bowden, Ross2, Doliskani, Javad3, Fouotsa, Tako Boris4, Galbraith, Steven D5, Kunzweiler, Sabrina6, Merz, Simon-Philipp7, Petit, Christophe8,9, Smith, Benjamin10, Stange, Katherine E11 kstange@math.colorado.edu, Ti, Yan Bo12, Vincent, Christelle13, Voloch, José Felipe14, Weitkämper, Charlotte9, Zobernig, Lukas5
Source: Computer Journal. Aug2024, Vol. 67 Issue 8, p2702-2719. 18p.
Subjects: Cryptography, Hashing, Random walks, Elliptic curves, Polynomials
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 |$\ell $| -isogeny graph, which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd's of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces, and applying Kummer surfaces and (v) using quantum random walks. [ABSTRACT FROM AUTHOR]
Copyright of Computer Journal is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Failing to Hash Into Supersingular Isogeny Graphs.
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  Data: An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of 'hard supersingular curves' that is equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular |$\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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computer Journal is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1093/comjnl/bxae038
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        Text: English
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        PageCount: 18
        StartPage: 2702
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      – SubjectFull: Cryptography
        Type: general
      – SubjectFull: Hashing
        Type: general
      – SubjectFull: Random walks
        Type: general
      – SubjectFull: Elliptic curves
        Type: general
      – SubjectFull: Polynomials
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      – TitleFull: Failing to Hash Into Supersingular Isogeny Graphs.
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              Text: Aug2024
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