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Efficient pairing computation on supersingular Abelian varieties
2007
Designs, Codes and Cryptography
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We present a general technique for the efficient computation of pairings on supersingular Abelian varieties. ...
This formulation, which we call the eta pairing, generalises results of Duursma and Lee for computing the Tate pairing on supersingular elliptic curves in characteristic three. ...
Abelian variety with embedding degree 3 × 4 = 12. ...
doi:10.1007/s10623-006-9033-6
fatcat:ofx6opm77balnjv5cmes6zdn4i
Using Abelian Varieties to Improve Pairing-Based Cryptography
2008
Journal of Cryptology
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We construct examples of good supersingular abelian varieties to use in pairing-based cryptography. ...
It is a theorem that the group law on every abelian variety is abelian. From now on, when we say abelian variety we mean abelian variety of dimension ≥ 1. Definition 2.1. ...
One could view E r as an abelian variety of dimension ϕ(r) and do pairing-based cryptography for that abelian variety (if one can compute the pairings). ...
doi:10.1007/s00145-008-9022-1
fatcat:ixmje3vbtzgfnihehyilj3pdui
Supersingular Abelian Varieties in Cryptology
[chapter]
2002
Lecture Notes in Computer Science
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Supersingular abelian varieties are natural candidates for these applications. ...
gives constructions of supersingular abelian varieties that are optimal for use in cryptography. ...
k not too large, in order to be able to compute the pairing efficiently. ...
doi:10.1007/3-540-45708-9_22
fatcat:mohd5mf3rbaudd4v6ea267abie
Pairings on hyperelliptic curves
[article]
2009
arXiv
pre-print
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We discuss the techniques used to optimize the pairing computation on hyperelliptic curves, and present many directions for further research. ...
We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairing-based cryptography. ...
Barreto, Anja Becker, Felix Fontein, Steven Galbraith, and Alfred Menezes for helpful discussions and comments on an earlier draft of the paper. ...
arXiv:0908.3731v2
fatcat:6nnfdtdi2rgnzpeyk642nf3mum
The supersingular isogeny problem in genus 2 and beyond
[article]
2020
arXiv
pre-print
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Let A/F_p and A'/F_p be supersingular principally polarized abelian varieties of dimension g>1. ...
. , ℓ)-isogeny graph in O(p^g-1) group operations on a classical computer, and O(√(p^g-1)) calls to the Grover oracle on a quantum computer. ...
Superspecial abelian varieties are isomorphic to a product of supersingular elliptic curves, though generally only as unpolarized abelian varieties. ...
arXiv:1912.00701v2
fatcat:qcuwlrsrnbfj7orcknghckhboe
Distortion maps for supersingular genus two curves
2009
Journal of Mathematical Cryptology
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Distortion maps are a useful tool for pairing based cryptography. ...
In this paper, we prove that distortion maps always exist for supersingular curves of genus g > 1. ...
Efficient pairing computation on these curves over extension fields F p m was studied by Duursma and Lee [5] . ...
doi:10.1515/jmc.2009.001
fatcat:cd7qi2pyyjhxnmax64yzhwpyiy
Supersingular Curves in Cryptography
[chapter]
2001
Lecture Notes in Computer Science
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Bounds on the possible values for k in the case of supersingular curves are given which imply that supersingular curves are weaker than the general case for cryptography. ...
In the elliptic curve case it was shown by Menezes, Okamoto and Vanstone that for supersingular curves one has k ≤ 6. In this paper curves of higher genus are studied. ...
about hyperelliptic curves in characteristic two; and Alice Silverberg for helpful comments on an earlier version of the paper. ...
doi:10.1007/3-540-45682-1_29
fatcat:642pifhyvvb6pjkttoxhj6xsti
Distortion maps for genus two curves
[article]
2006
arXiv
pre-print
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Distortion maps are a useful tool for pairing based cryptography. ...
In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12). ...
Finally, a curve C is called supersingular if Jac(C) is a supersingular abelian variety. Let r > 2 be a prime dividing q k − 1, and coprime to q. ...
arXiv:math/0611471v1
fatcat:y3tudjceajhbngy2w2adcuzdeq
Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves
2020
Journal of Mathematical Cryptology
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What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian ...
Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. ...
Acknowledgement: We thank the American Institute of Mathematics (AIM) for supporting a workshop on multilinear maps where the initial seeds for this work were developed, and the Banff International Research ...
doi:10.1515/jmc-2015-0047
fatcat:hj6dt3wobvf5xktfzc7qd5pj4a
Failing to hash into supersingular isogeny graphs
[article]
2022
arXiv
pre-print
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for which computing the endomorphism ring is as difficult as it is for random supersingular curves. ...
to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces; and (v) using quantum random walks. ...
Oort [43] has shown that every superspecial abelian surface is isomorphic as an unpolarized abelian variety to a product of supersingular elliptic curves, and that every supersingular abelian surface ...
arXiv:2205.00135v1
fatcat:e3dpuq3cpbhexj46rhjivxs6vu
Supersingular hyperelliptic curves of genus 2 over finite fields
2005
Applied Mathematics and Computation
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In this paper we describe an elementary criterion how to determine supersingular hyperelliptic curves of genus 2, directly using only the given Weierstrass equation. ...
Embedding degree of supersingular abelian variety of dimension 2 over F p In this section, we focus more on the supersingular abelian varieties A over F p . ...
Ã
Main theorem There are criterions to check whether or not abelian variety A is supersingular, once its characteristic polynomial is computed. ...
doi:10.1016/j.amc.2004.03.030
fatcat:as2ie2jlefbtffbtae3kzii5rq
Zeta function and cryptographic exponent of supersingular curves of genus 2
[article]
2007
arXiv
pre-print
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We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. ...
As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves. ...
on supersingular abelian varieties over a finite field k. ...
arXiv:0704.1951v1
fatcat:b7743wlcijgxjmzs7hbyc37rkm
Construction of Hyperelliptic Curves with CM and Its Application to Cryptosystems
[chapter]
2000
Lecture Notes in Computer Science
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Construction of secure hyperelliptic curves is of most important yet most difficult problem in design of cryptosystems based on the discrete logarithm problems on hyperelliptic curves. ...
As we show in this paper, it costs computation time of an exponential function in the discriminant of the CM field. ...
Gerhard Frey for interesting comments on [33] and Dr. Michael Müller for sending us a copy of Dr. Spallek's thesis. ...
doi:10.1007/3-540-44448-3_20
fatcat:z2gwjbvgpzdhpe6xedky7z4o5q
On the Security of Pairing-Friendly Abelian Varieties over Non-prime Fields
[chapter]
2009
Lecture Notes in Computer Science
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This result could drastically reduce the security of pairing based systems, in which the elliptic curve discrete logarithm problem (ECDLP) on a curve defined over some field is mapped to the discrete logarithm ...
In this paper it will be shown that the supersingular curves can be chosen in such a way that the minimal embedding field is in fact F q k and that Hitt's result does not in fact apply to the optimal supersingular ...
This has large security implications for pairing based systems as the DLP is more efficiently computed in the smaller field and hence the security of the ECDLP and the DLP in such a case would not be equivalently ...
doi:10.1007/978-3-642-03298-1_4
fatcat:gpv2yc4ln5fcnosyvounarc6eq
Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves
[article]
2018
arXiv
pre-print
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What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian ...
Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. ...
Acknowledgments We thank the American Institute of Mathematics (AIM) for supporting a workshop on multilinear maps where the initial seeds for this work were developed, and the Banff International Research ...
arXiv:1807.03038v3
fatcat:brihvmwo7jgrbih2br2dcqaz7u
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