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Supersingular Isogeny Graphs and Endomorphism Rings: Reductions and Solutions [chapter]

Kirsten Eisenträger, Sean Hallgren, Kristin Lauter, Travis Morrison, Christophe Petit
2018 Lecture Notes in Computer Science  
We prove reductions between the problem of path finding in the -isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism  ...  In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings.  ...  We thank John Voight for many helpful discussions regarding orders in quaternion algebras and their connection with supersingular elliptic curves.  ... 
doi:10.1007/978-3-319-78372-7_11 fatcat:wvs4ee46hbak3plxmpmmlnqsmm

The supersingular isogeny path and endomorphism ring problems are equivalent [article]

Benjamin Wesolowski
2021 arXiv   pre-print
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  ...  The presumed hardness of these problems is foundational for isogeny-based cryptography.  ...  As an application 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  ... 
arXiv:2111.01481v1 fatcat:l56uyhwrmzbv7gxqhmnme4mx3a

The supersingular isogeny path and endomorphism ring problems are equivalent

Benjamin Wesolowski
2022 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)  
We prove that the path-finding problem in -isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised  ...  The presumed hardness of these problems is foundational for isogeny-based cryptography.  ...  As an application we prove that the path-finding problem in -isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time  ... 
doi:10.1109/focs52979.2021.00109 fatcat:uxtllin6djfizn3qsmgydudhm4

Computing isogenies between supersingular elliptic curves over F_p [article]

Christina Delfs, Steven D. Galbraith
2013 arXiv   pre-print
Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p. We want to construct an isogeny phi: E --> E'.  ...  In this paper we consider the structure of the isogeny graph of supersingular elliptic curves over F_p.  ...  Acknowledgements We thank David Kohel and Drew Sutherland for helpful conversations and Marco Streng for the idea of the proof of Proposition 2.6.  ... 
arXiv:1310.7789v1 fatcat:pvbpmuwzrvgoncxbqfzaje72nm

Computational problems in supersingular elliptic curve isogenies

Steven D. Galbraith, Frederik Vercauteren
2018 Quantum Information Processing  
We give a brief survey of elliptic curve isogenies and the computational problems relevant for supersingular isogeny crypto.  ...  Supersingular isogeny cryptography is attracting attention due to the fact that there are no quantum attacks known against it that are significantly faster than classical attacks.  ...  Acknowledgement We thank Sean Hallgren, Christophe Petit and Drew Sutherland for discussions and comments.  ... 
doi:10.1007/s11128-018-2023-6 fatcat:7zlqrtmefffapbfax5qqwzc42u

Higher-degree supersingular group actions [article]

Mathilde Chenu
2021 arXiv   pre-print
We investigate the isogeny graphs of supersingular elliptic curves over 𝔽_p^2 equipped with a d-isogeny to their Galois conjugate.  ...  We investigate constructive and destructive aspects of these graphs in isogeny-based cryptography, including generalizations of the CSIDH cryptosystem and the Delfs-Galbraith algorithm.  ...  F -endomorphism rings of these curves are an imaginary quadratic ring, and the ideal class group of this ring has a convenient and efficiently-computable commutative action on the F -subgraph.  ... 
arXiv:2107.08832v1 fatcat:gt2tkvnnprdmrfsupvtcl6sbya

Orienting supersingular isogeny graphs

Leonardo Colò, David Kohel
2020 Journal of Mathematical Cryptology  
AbstractWe introduce a category of 𝓞-oriented supersingular elliptic curves and derive properties of the associated oriented and nonoriented ℓ-isogeny supersingular isogeny graphs.  ...  As an application we introduce an oriented supersingular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative  ...  We study some theoretical and practical aspects of the endomorphism ring of a supersingular elliptic curve and their connection with isogeny graphs.  ... 
doi:10.1515/jmc-2019-0034 fatcat:ecrikykih5e45pf46q4kpxm2iq

Orienteering with one endomorphism [article]

Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, Ha T. N. Tran
2022 arXiv   pre-print
We use the theory of oriented supersingular isogeny graphs and algorithms for taking ascending/descending/horizontal steps on such graphs.  ...  In supersingular isogeny-based cryptography, the path-finding problem reduces to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism?  ...  Central are the path-finding problem (to find a path between two specified elliptic curves in a supersingular -isogeny graph), and the endomorphism ring problem (to compute the endomorphism ring of a supersingular  ... 
arXiv:2201.11079v2 fatcat:hmlhxcrp4jfh5b4vnnojcldg4e

Supersingular Curves With Small Non-integer Endomorphisms [article]

Jonathan Love, Dan Boneh
2020 arXiv   pre-print
searching on ℓ-isogeny graphs.  ...  We introduce a special class of supersingular curves over F_p^2, characterized by the existence of non-integer endomorphisms of small degree. A number of properties of this set is proved.  ...  We would like to thank John Voight for fruitful discussion without which we would not have found the algorithms in Section 7, and Akshay Venkatesh for pointing us towards the key ideas in Section 5.  ... 
arXiv:1910.03180v3 fatcat:hhyzslzqpvg35ftz2rdfkl5elm

Failing to hash into supersingular isogeny graphs [article]

Jeremy 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 (+3 others)
2022 arXiv   pre-print
Or, even better, to produce a hash function to the vertices of the supersingular ℓ-isogeny graph which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring.  ...  for which computing the endomorphism ring is as difficult as it is for random supersingular curves.  ...  The supersingular endomorphism ring problem and the supersingular isogeny problem are related [26, 31, 54] .  ... 
arXiv:2205.00135v1 fatcat:e3dpuq3cpbhexj46rhjivxs6vu

On oriented supersingular elliptic curves [article]

Hiroshi Onuki
2020 arXiv   pre-print
We revisit theoretical background on OSIDH, that is an isogeny-based key-exchange protocol proposed by Col\'o and Kohel at NutMiC 2019. We give a proof of a fundamental theorem for OSIDH.  ...  The theorem was stated by Col\'o and Kohel without proof.  ...  The endomorphism ring of a supersingular elliptic curve is isomorphic to a maximal order of a quaternion algebra. The isogeny problem is closely related the structure of the endomorphism ring.  ... 
arXiv:2002.09894v3 fatcat:kdw7ne4fanc2dcn3sp47smftyy

Supersingular curves with small noninteger endomorphisms

Jonathan Love, Dan Boneh
2020 The Open Book Series  
LOVE AND DAN BONEH (b) The endomorphism rings of M-small supersingular curves, and isogenies between any two of them, can heuristically be computed in time polynomial in M and log p (Section 7).  ...  found by searching on -isogeny graphs.  ...  and constructive feedback.  ... 
doi:10.2140/obs.2020.4.7 fatcat:5kzqtmlfe5aztg2kodiiffiqai

Cryptographic Hash Functions from Expander Graphs

Denis X. Charles, Kristin E. Lauter, Eyal Z. Goren
2007 Journal of Cryptology  
from hardness of computing isogenies between supersingular elliptic curves.  ...  We estimate the cost per bit to compute these hash functions, and we implement our hash function for several members of the Pizer and LPS graph families and give actual timings.  ...  We claim that the graph G(p, ) for ≡ 3 (mod 4) cannot have small cycles starting from any vertex representing a supersingular elliptic curve with endomorphism ring m.  ... 
doi:10.1007/s00145-007-9002-x fatcat:aeu37cl6fvhmpdlrc5pu22wp7e

Verifiable Delay Functions from Supersingular Isogenies and Pairings [chapter]

Luca De Feo, Simon Masson, Christophe Petit, Antonio Sanso
2019 Lecture Notes in Computer Science  
We discuss both the advantages and drawbacks of our constructions, we study their security and we demonstrate their practicality with a proof-of-concept implementation.  ...  We are grateful to the anonymous reviewers for their attentive reading and their helpful comments.  ...  Barbulescu, Jeff Burdges, Wouter Castryck, Jeroen Demeyer, Andreas Enge, Steven Galbraith, Matthew Green, Philipp Jovanovic, Jean Kieffer, Enea Milio, Aurel Page, Lorenz Panny, Damien Robert, Barak Shani and  ... 
doi:10.1007/978-3-030-34578-5_10 fatcat:xz5fbejjdngk7glu5vp7w5gp44

Mathematics of Isogeny Based Cryptography [article]

Luca De Feo
2017 arXiv   pre-print
The reader will excuse us for the inaccuracies and the omissions.  ...  They try to provide a guide for Masters' students to get through the vast literature on elliptic curves, without getting lost on their way to learning isogeny based cryptography.  ...  The endomorphism ring We have already defined an endomorphism as an isogeny from a curve to itself.  ... 
arXiv:1711.04062v1 fatcat:i5l6qyzxezhxxbj3a2cee3klwm
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