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

2009
*
arXiv
*
pre-print

We describe

arXiv:0806.2995v2
fatcat:xpkzor6lcbdadekeo4fvwgwil4
*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%). ...##
###
Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,

2009
*
Journal of Cryptology
*

We describe

doi:10.1007/s00145-009-9038-1
fatcat:jwsrx5kdvnavte6tfelyvwe53a
*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. ...##
###
Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves
[chapter]

*
Advances in Cryptology – EUROCRYPT 2008
*

We describe

doi:10.1007/978-3-540-78967-3_10
dblp:conf/eurocrypt/Smith08
fatcat:yl3marhndjcslpyqdswlmeooqu
*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. ...##
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Efficient construction of secure hyperelliptic discrete logarithm problems
[chapter]

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. ...

##
###
A Survey Report On Elliptic Curve Cryptography

2011
*
International Journal of Electrical and Computer Engineering (IJECE)
*

This paper also discuss

doi:10.11591/ijece.v1i2.86
fatcat:etikame46ve2nkrio52rprs7ky
*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*genus*≥*3**the*generic*curve*is not*hyperelliptic*.*Hyperelliptic**curves*can be used*in*cryptosystems based on*the*discerete*logarithm**problem*[40] . ...##
###
Hyperelliptic Curves for the Vector Decomposition Problem over Fields of Even Characteristic

2015
*
Journal of Applied Mathematics
*

We present an infinite family

doi:10.1155/2015/197097
fatcat:nbbym7ywind6zbh4ybh3vbtzhe
*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. ...##
###
Curves, Jacobians, and Cryptography
[article]

2018
*
arXiv
*
pre-print

index calculus

arXiv:1807.05270v2
fatcat:c76zy3njpzgkzazgaapigkdr3q
*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**genus*≤*3*provide candidates for secure crypto systems based on*discrete**logarithms*. ...##
###
Cover and Decomposition Index Calculus on Elliptic Curves Made Practical
[chapter]

2012
*
Lecture Notes in Computer Science
*

We present a new "cover

doi:10.1007/978-3-642-29011-4_3
fatcat:23ppmsbcyrgf7atw55jgrbgztq
*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. ...##
###
Correspondences on Hyperelliptic Curves and Applications to the Discrete Logarithm
[chapter]

2012
*
Lecture Notes in Computer Science
*

An important observation

doi:10.1007/978-3-642-25261-7_1
fatcat:jchxmgjarbaldk5rvo2uas7yri
*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. ...##
###
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
[article]

2017
*
arXiv
*
pre-print

We use these graphs, together with a recent algorithm

arXiv:1506.00522v2
fatcat:l6tthtqs7zdv7evkw2bpitso6i
*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. ...##
###
Computational aspects of curves of genus at least 2
[chapter]

1996
*
Lecture Notes in Computer Science
*

This survey discusses algorithms

doi:10.1007/3-540-61581-4_63
fatcat:jhppxzsferfhvnxepkrz43jc5i
*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! ...##
###
Constructive and destructive facets of Weil descent on elliptic curves

2002
*
Journal of Cryptology
*

On

doi:10.1007/s00145-001-0011-x
fatcat:slnfxmjt4fb5tmucrv5vusg6tm
*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 ...##
###
Extending the GLS endomorphism to speed up GHS Weil descent using Magma

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. ...

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

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 ...

##
###
Discrete logarithms in curves over finite fields
[article]

2007
*
arXiv
*
pre-print

A survey on algorithms for computing

arXiv:0712.3916v1
fatcat:yt776ztyrbacdi3rauevpcwjka
*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. ...
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