A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2012; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

##
###
Computational algebra and algebraic curves

2003
*
ACM SIGSAM Bulletin
*

In this survey, we briefly describe some open problems related to algebraic

doi:10.1145/968708.968713
fatcat:w6a6yiof4fh4jjldlzplhbcm4q
*curves*which can be approached from*a**computational*viewpoint. ... The development*of**computational*techniques in the last decade has made possible to attack some classical problems*of*algebraic geometry. ... Acknowledgments Most*of*the topics discussed in this paper are joint work*with*my collaborators. I would like to thank J. Gutierrez ...##
###
Computational algebra and algebraic curves
[article]

2006
*
arXiv
*
pre-print

In this survey, we briefly describe some open problems related to algebraic

arXiv:math/0601398v1
fatcat:ncta4nxdqngozniaqbulonfg6q
*curves*which can be approached from*a**computational*viewpoint. ... The development*of**computational*techniques in the last decade has made possible to attack some classical problems*of*algebraic geometry. ... The*families**of**hyperelliptic**curves**with*reduced*automorphism*group (i.e., the*automorphism*group modulo the*hyperelliptic*involution) isomorphic to*A*4 or*a*cyclic group, are studied in [Sh6] . ...##
###
Speeding Up Point Multiplication on Hyperelliptic Curves with Efficiently-Computable Endomorphisms
[chapter]

2002
*
Lecture Notes in Computer Science
*

For this special

doi:10.1007/3-540-46035-7_13
fatcat:qaxk4acdxbhnnmi43g6kmzpnie
*family**of**curves*,*a*speedup*of*up to 55 (59) % can be achieved over the best general methods for*a*160-bit point multiplication in case*of**genus*g =*2*(3). ... So the extended method for speeding point multiplication applies to*a*much larger*family**of**hyperelliptic**curves*over finite fields that have*efficiently*-*computable*endomorphisms. ...*Hyperelliptic**curves**with**efficient*endomorphisms In this Section we first collect*a**family**of**hyperelliptic**curves**of**genus*g over F q that have*efficiently*-*computable*endomorphisms φ. ...##
###
Pairings on hyperelliptic curves
[article]

2009
*
arXiv
*
pre-print

We assemble and reorganize the recent work in the area

arXiv:0908.3731v2
fatcat:6nnfdtdi2rgnzpeyk642nf3mum
*of**hyperelliptic**pairings*: We survey the research*on*constructing*hyperelliptic**curves*suitable for*pairing*-based cryptography. ... We discuss the techniques used to optimize the*pairing**computation**on**hyperelliptic**curves*, and present many directions for further research. ...*pairings*. ...##
###
Hyperelliptic Curves with Extra Involutions

2005
*
LMS Journal of Computation and Mathematics
*

AbstractThe purpose

doi:10.1112/s1461157000000917
fatcat:djfljrsq5jdc7hwg6o5bqv6rhy
*of*this paper is to study*hyperelliptic**curves**with*extra involutions. ... The locusLgof such*genus*-ghyperelliptic*curves*is ag-dimensional subvariety*of*the moduli space*of**hyperelliptic*curvesHg. ... Shimura's*family*and Earle's*family**of**curves*(that is,*with*non-trivial obstruction) are both*families**of**hyperelliptic**curves**with**automorphism*group*of*order*2*; see [9] . ...##
###
Zeta function and cryptographic exponent of supersingular curves of genus 2
[article]

2007
*
arXiv
*
pre-print

We

arXiv:0704.1951v1
fatcat:b7743wlcijgxjmzs7hbyc37rkm
*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*. ... We display these*computations*in an appendix where we select*a**family**of*representatives*of*all these*curves*up to geometric isomorphism and we exhibit equations and the zeta function*of*all their twists ... It is*a*pleasure to thank Christophe Ritzenthaler for his help in finding some*of*the equations*of*the twisted*curves*. ...##
###
Algebraic curves and cryptography

2005
*
Finite Fields and Their Applications
*

*A*

*hyperelliptic*

*curve*C

*of*

*genus*g 1 over K can be defined by

*a*non-singular equation

*of*the form where f, h ∈ K[x], f is monic, deg f = 2g + 1, and deg h g. ... In particular, we describe some

*families*

*of*

*curves*whose DLP is easier than the general case. Protocols using bilinear

*pairings*and some related

*computational*problems are covered in §5. ... Acknowledgments We are grateful to Claus Diem for his extensive comments

*on*an earlier draft

*of*the paper, and the two anonymous referees for their useful remarks. ...

##
###
Construction of Hyperelliptic Curves with CM and Its Application to Cryptosystems
[chapter]

2000
*
Lecture Notes in Computer Science
*

Construction

doi:10.1007/3-540-44448-3_20
fatcat:z2gwjbvgpzdhpe6xedky7z4o5q
*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*. ... This paper presents new algorithms to find explicit models*of**hyperelliptic**curves**with*CM. ... Gerhard Frey for interesting comments*on*[33] and Dr. Michael Müller for sending us*a*copy*of*Dr. Spallek's thesis. ...##
###
Computational aspects of curves of genus at least 2
[chapter]

1996
*
Lecture Notes in Computer Science
*

This survey discusses algorithms and explicit calculations for

doi:10.1007/3-540-61581-4_63
fatcat:jhppxzsferfhvnxepkrz43jc5i
*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. ... But I thank especially Noam Elkies, for many insightful comments*on*an earlier draft*of*this survey. ...##
###
Supersingular Curves in Cryptography
[chapter]

2001
*
Lecture Notes in Computer Science
*

In the elliptic

doi:10.1007/3-540-45682-1_29
fatcat:642pifhyvvb6pjkttoxhj6xsti
*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. ... 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. ... about*hyperelliptic**curves*in characteristic two; and Alice Silverberg for helpful comments*on*an earlier version*of*the paper. ...##
###
Optimal Eta Pairing on Supersingular Genus-2 Binary Hyperelliptic Curves
[chapter]

2012
*
Lecture Notes in Computer Science
*

As

doi:10.1007/978-3-642-27954-6_7
fatcat:bmcheryyl5fgpd3pmjgw476gn4
*a*proof*of*concept, we detail an optimized software implementation and an FPGA accelerator for*computing*the proposed optimal Eta*pairing**on**a**genus*-*2**hyperelliptic**curve*over F*2*367 , which satisfies ... This article presents*a*novel*pairing*algorithm over supersingular*genus*-*2*binary*hyperelliptic**curves*. ... First*of*all, the authors would like to express their deepest thanks to Guillaume Hanrot who advised us to have*a*go at*genus*-*2**pairings*. He shall receive here our utmost gratitude. ...##
###
Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

2015
*
Mathematics of Computation
*

If it vanishes, then the use

doi:10.1090/mcom3032
fatcat:gbsk56ddofaqbhhutu444wf5di
*of*these invariants also allows the explicit determination*of**a*model over the field*of*moduli; if not, then*one*obtains*a**hyperelliptic*model over*a*degree*2*extension*of*... This paper is devoted to the explicit description*of*the Galois descent obstruction for*hyperelliptic**curves**of*arbitrary*genus*whose reduced*automorphism*group is cyclic*of*order coprime to the characteristic ... for*genus*3*hyperelliptic**curves**with*great*efficiency*. ...##
###
On explicit descent of marked curves and maps

2016
*
Research in Number Theory
*

We give

doi:10.1007/s40993-016-0057-3
fatcat:q4rqwdi46zcuhfn3nmu3bpivdm
*a*constructive version*of*their results, based*on*an algebraic version*of*the notion*of*branches*of**a*morphism and allowing us to extend the aforementioned results to the wildly ramified case. ... Classical criteria due to D\'ebes and Emsalem can be used to prove this statement in the presence*of**a*smooth point, and in fact these results imply more generally that*a*marked*curve*descends to its field ... We would like to thank Bryan Birch, Enric Nart, Andrew Obus, and the anonymous referee for their valuable comments*on*previous versions*of*this article. ...##
###
Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three

2007
*
Journal of Cryptology
*

We present

doi:10.1007/s00145-007-9014-6
fatcat:clczgl7xr5b5vgekyvma25fxga
*a*heuristic analysis*of*the algorithm which indicates that the DLP in degree 0 class groups*of*non-*hyperelliptic**curves**of**genus*3 can be solved in an expected time ofÕ(q). ... We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups*of*non-*hyperelliptic**curves**of**genus*3 over finite fields. ... Acknowledgments It is*a*great pleasure to thank G. Frey, P. Gaudry, F. Heß, R. van der Hofstad, W. König, K. Magaard, N. Thériault and E. Viehweg for discussions and helpful comments. ...##
###
The arithmetic of genus two curves
[article]

2012
*
arXiv
*
pre-print

and the arithmetic

arXiv:1209.0439v3
fatcat:prlfb6uutjbexc2vl6ajwq7l44
*on*the Jacobians*of**genus**2*, and their applications to cryptography. ... Choosing*genus**2**curves*suitable for such applications is an important step*of*such algorithms. ... For*a*fixed group G*one*can*compute*the locus*of**genus*g*hyperelliptic**curves**with**automorphism*group G. For*genus**2*this loci is well described as subvarieties*of*M*2*. ...
« Previous

*Showing results 1 — 15 out of 216 results*