Filters








335 Hits in 3.0 sec

Efficient pairing computation on supersingular Abelian varieties

Paulo S. L. M. Barreto, Steven D. Galbraith, Colm Ó' hÉigeartaigh, Michael Scott
2007 Designs, Codes and Cryptography  
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

K. Rubin, A. Silverberg
2008 Journal of Cryptology  
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]

Karl Rubin, Alice Silverberg
2002 Lecture Notes in Computer Science  
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]

Jennifer Balakrishnan, Juliana Belding, Sarah Chisholm, Kirsten Eisentraeger, Katherine Stange, Edlyn Teske
2009 arXiv   pre-print
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]

Craig Costello, Benjamin Smith
2020 arXiv   pre-print
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

Steven D. Galbraith, Jordi Pujolàs, Christophe Ritzenthaler, Benjamin Smith
2009 Journal of Mathematical Cryptology  
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]

Steven D. Galbraith
2001 Lecture Notes in Computer Science  
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]

Steven D. Galbraith, Jordi Pujolàs, Christophe Ritzenthaler and Benjamin Smith
2006 arXiv   pre-print
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

Dan Boneh, Darren Glass, Daniel Krashen, Kristin Lauter, Shahed Sharif, Alice Silverberg, Mehdi Tibouchi, Mark Zhandry
2020 Journal of Mathematical Cryptology  
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]

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
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

Young Ju Choie, Eun Kyung Jeong, Eun Jeong Lee
2005 Applied Mathematics and Computation  
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]

Gabriel Cardona, Enric Nart
2007 arXiv   pre-print
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]

Jinhui Chao, Kazuto Matsuo, Hiroto Kawashiro, Shigeo Tsujii
2000 Lecture Notes in Computer Science  
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]

Naomi Benger, Manuel Charlemagne, David Mandell Freeman
2009 Lecture Notes in Computer Science  
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]

Dan Boneh and Darren Glass and Daniel Krashen and Kristin Lauter and Shahed Sharif and Alice Silverberg and Mehdi Tibouchi and Mark Zhandry
2018 arXiv   pre-print
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
« Previous Showing results 1 — 15 out of 335 results