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Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves [article]

Benjamin Smith
2009 arXiv   pre-print
We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves,  ...  These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic p > 3.  ...  In particular, we could move a discrete logarithm problem for 18.58% of these curves (recall that Theorem 2 predicts a success rate of about 18.57%).  ... 
arXiv:0806.2995v2 fatcat:xpkzor6lcbdadekeo4fvwgwil4

Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,

Benjamin Smith
2009 Journal of Cryptology  
We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, which are vulnerable  ...  Using Recillas' trigonal construction [12] , A may be realized as the Jacobian of a genus 3 curve X.  ...  for explicit isogenies of general Jacobians of genus 3 hyperelliptic curves other than the one presented here.  ... 
doi:10.1007/s00145-009-9038-1 fatcat:jwsrx5kdvnavte6tfelyvwe53a

Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves [chapter]

Benjamin Smith
Advances in Cryptology – EUROCRYPT 2008  
We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where  ...  We provide explicit formulae for isogenies with kernel isomorphic to (Z/2Z) 3 (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3  ...  Acknowledgements This work was supported by EPSRC grant EP/C014839/1, and a large part of it was completed in the Department of Mathematics at Royal Holloway, University of London.  ... 
doi:10.1007/978-3-540-78967-3_10 dblp:conf/eurocrypt/Smith08 fatcat:yl3marhndjcslpyqdswlmeooqu

Efficient construction of secure hyperelliptic discrete logarithm problems [chapter]

Jinhui Chao, Nori Matsuda, Shigeo Tsujii
1997 Lecture Notes in Computer Science  
In this paper, ecient algorithms are presented to construct secure discrete logarithm problems on hyperelliptic curves whose Jacobian varieties are either simple or isogenous to a product of simple abelian  ...  Hyperelliptic curves have been used to dene discrete logarithm problems as cryptographic one-way functions.  ...  Acknowledgment: The authors wish to thank Prof. Fumiyuki Momose of Dept. of Math. Chuo University, Japan for helpful discussions.  ... 
doi:10.1007/bfb0028485 fatcat:qrm2dgwxzfeepafsiwmzkqly2u

A Survey Report On Elliptic Curve Cryptography

Samta Gajbhiye, Monisha Sharma, Samir Dashputre
2011 International Journal of Electrical and Computer Engineering (IJECE)  
This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems.  ...  It also explains how isogenenies between elliptic curve provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons.  ...  All curves of genus 2 are hyperelliptic, but for genus3 the generic curve is not hyperelliptic. Hyperelliptic curves can be used in cryptosystems based on the discerete logarithm problem [40] .  ... 
doi:10.11591/ijece.v1i2.86 fatcat:etikame46ve2nkrio52rprs7ky

Hyperelliptic Curves for the Vector Decomposition Problem over Fields of Even Characteristic

Seungkook Park
2015 Journal of Applied Mathematics  
We present an infinite family of hyperelliptic curves of genus two over a finite field of even characteristic which are suitable for the vector decomposition problem.  ...  Acknowledgment This research was supported by the Sookmyung Women's University Research Grants (1-1103-0682) .  ...  Thus the elliptic curve discrete logarithm problem (ECDLP), and hence the CDHP on the one-dimensional subspace, is vulnerable to the MOV attack.  ... 
doi:10.1155/2015/197097 fatcat:nbbym7ywind6zbh4ybh3vbtzhe

Curves, Jacobians, and Cryptography [article]

Gerhard Frey, Tony Shaska
2018 arXiv   pre-print
index calculus in Picard groups, isogenies of Jacobians via correspondences and applications to discrete logarithms.  ...  In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and 3 cryptography, including Diffie-Hellman Key Exchange,  ...  As consequence one sees that only elliptic and hyperelliptic curves of genus3 provide candidates for secure crypto systems based on discrete logarithms.  ... 
arXiv:1807.05270v2 fatcat:c76zy3njpzgkzazgaapigkdr3q

Cover and Decomposition Index Calculus on Elliptic Curves Made Practical [chapter]

Antoine Joux, Vanessa Vitse
2012 Lecture Notes in Computer Science  
We present a new "cover and decomposition" attack on the elliptic curve discrete logarithm problem, that combines Weil descent and decomposition-based index calculus into a single discrete logarithm algorithm  ...  We give a real-size example of discrete logarithm computations on a curve over a 151-bit degree 6 extension field, which would not have been practically attackable using previously known algorithms.  ...  We acknowledge that the results in this paper have been achieved using the PRACE Research Infrastructure resource Curie based in France at TGCC, Bruyères-le-Chatel.  ... 
doi:10.1007/978-3-642-29011-4_3 fatcat:23ppmsbcyrgf7atw55jgrbgztq

Correspondences on Hyperelliptic Curves and Applications to the Discrete Logarithm [chapter]

Gerhard Frey, Ernst Kani
2012 Lecture Notes in Computer Science  
An important observation of Smith [S] is that for "many" hyperelliptic curves of genus 3 there is an explicit isogeny of their Jacobian variety to the Jacobian of a non-hyperelliptic curve.  ...  Because of index-calculus algorithms one has to avoid curves of genus ≥ 4 and non-hyperelliptic curves of genus 3.  ...  of genus 3 and so make the discrete logarithm insecure again.  ... 
doi:10.1007/978-3-642-25261-7_1 fatcat:jchxmgjarbaldk5rvo2uas7yri

Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem [article]

Dimitar Jetchev, Benjamin Wesolowski
2017 arXiv   pre-print
We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and Robert for computing explicit isogenies in genus 2, to prove random self-reducibility of the discrete logarithm problem within  ...  In addition, we remove the heuristics in the complexity analysis of an algorithm of Galbraith for explicitly computing isogenies between two elliptic curves in the same isogeny class, and extend it to  ...  Acknowledgements We thank Emmanuel Kowalski, Philippe Michel, Ken Ribet and Damien Robert for useful conversations. The first author was supported by the Swiss National Science Foundation.  ... 
arXiv:1506.00522v2 fatcat:l6tthtqs7zdv7evkw2bpitso6i

Computational aspects of curves of genus at least 2 [chapter]

Bjorn Poonen
1996 Lecture Notes in Computer Science  
This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields.  ...  Miscellaneous examples and a list of possible future projects are given at the end.  ...  Acknowledgements An enormous number of people have helped me gather an enormous number of references!  ... 
doi:10.1007/3-540-61581-4_63 fatcat:jhppxzsferfhvnxepkrz43jc5i

Constructive and destructive facets of Weil descent on elliptic curves

P. Gaudry, F. Hess, N. P. Smart
2002 Journal of Cryptology  
On the other hand, we show that the same technique may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic  ...  In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a nite eld of characteristic two of  ...  There is an index calculus style algorithm to solve the hyperelliptic discrete logarithm problem in a hyperelliptic curve of genus g over the eld F q which requires a factor base of size O(q) and which  ... 
doi:10.1007/s00145-001-0011-x fatcat:slnfxmjt4fb5tmucrv5vusg6tm

Extending the GLS endomorphism to speed up GHS Weil descent using Magma

Jesús-Javier Chi-Domínguez, Francisco Rodríguez-Henríquez, Benjamin Smith
2021 Finite Fields and Their Applications  
Jacobian J_H(𝔽_q) of the genus-g hyperelliptic curve H corresponding to the image of the GHS Weil-descent attack applied to E/𝔽_q^ℓ, and that this endomorphism yields a factor-n speedup when using standard  ...  A Magma implementation of our algorithm finds the aforementioned discrete logarithm in about 1,035 CPU-days.  ...  Acknowledgement The authors would like to acknowledge the anonymous referees whose comments and suggestions greatly helped us to improve the manuscript.  ... 
doi:10.1016/j.ffa.2021.101891 fatcat:3eknenospvhytc2yucrcq55cze

Genus 2 Hyperelliptic Curve Families with Explicit Jacobian Order Evaluation and Pairing-Friendly Constructions [chapter]

Aurore Guillevic, Damien Vergnaud
2013 Lecture Notes in Computer Science  
The use of elliptic and hyperelliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve.  ...  We extend and generalize Satoh's idea to provide explicit formulas for the zeta function of the Jacobian of genus 2 hyperelliptic curves of the form Y 2 = X 5 + aX 3 + bX and Y 2 = X 6 + aX 3 + b (with  ...  Acknowledgments This work was supported in part by the French ANR-09-VERS-016 BEST Project and by the Commission of the European Communities through the ICT program under contract ICT-2007-216676 ECRYPT  ... 
doi:10.1007/978-3-642-36334-4_16 fatcat:cdp3jvavxfhr3h37fmz4c5pjvi

Discrete logarithms in curves over finite fields [article]

Andreas Enge
2007 arXiv   pre-print
A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.  ...  In particular, in the case a = 3 and b = 4 the complexity is O(q); so the discrete logarithm problem in non-hyperelliptic C a,b curves of genus 3 is not harder than in hyperelliptic curves of genus 2 defined  ...  He explicitly computes an isogeny to a non-hyperelliptic curve of genus 3, which allows to transport the discrete logarithm problem and to solve it via the algorithm of Section 3.3.  ... 
arXiv:0712.3916v1 fatcat:yt776ztyrbacdi3rauevpcwjka
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