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Computational algebra and algebraic curves

Tanush Shaska
2003 ACM SIGSAM Bulletin  
In this survey, we briefly describe some open problems related to algebraic 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  ... 
doi:10.1145/968708.968713 fatcat:w6a6yiof4fh4jjldlzplhbcm4q

Computational algebra and algebraic curves [article]

Tanush Shaska
2006 arXiv   pre-print
In this survey, we briefly describe some open problems related to algebraic 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] .  ... 
arXiv:math/0601398v1 fatcat:ncta4nxdqngozniaqbulonfg6q

Speeding Up Point Multiplication on Hyperelliptic Curves with Efficiently-Computable Endomorphisms [chapter]

Young-Ho Park, Sangtae Jeong, Jongin Lim
2002 Lecture Notes in Computer Science  
For this special 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.  ... 
doi:10.1007/3-540-46035-7_13 fatcat:qaxk4acdxbhnnmi43g6kmzpnie

Pairings on hyperelliptic curves [article]

Jennifer Balakrishnan, Juliana Belding, Sarah Chisholm, Kirsten Eisentraeger, Katherine Stange, Edlyn Teske
2009 arXiv   pre-print
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.  ...  We discuss the techniques used to optimize the pairing computation on hyperelliptic curves, and present many directions for further research.  ...  pairings.  ... 
arXiv:0908.3731v2 fatcat:6nnfdtdi2rgnzpeyk642nf3mum

Hyperelliptic Curves with Extra Involutions

J. Gutierrez, T. Shaska
2005 LMS Journal of Computation and Mathematics  
AbstractThe purpose 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] .  ... 
doi:10.1112/s1461157000000917 fatcat:djfljrsq5jdc7hwg6o5bqv6rhy

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.  ...  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.  ... 
arXiv:0704.1951v1 fatcat:b7743wlcijgxjmzs7hbyc37rkm

Algebraic curves and cryptography

Steven Galbraith, Alfred Menezes
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.  ... 
doi:10.1016/j.ffa.2005.05.001 fatcat:o3oxj44uyfa33p3krkkhr2rrva

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.  ...  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.  ... 
doi:10.1007/3-540-44448-3_20 fatcat:z2gwjbvgpzdhpe6xedky7z4o5q

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.  ...  But I thank especially Noam Elkies, for many insightful comments on an earlier draft of this survey.  ... 
doi:10.1007/3-540-61581-4_63 fatcat:jhppxzsferfhvnxepkrz43jc5i

Supersingular Curves in Cryptography [chapter]

Steven D. Galbraith
2001 Lecture Notes in Computer Science  
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.  ...  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.  ... 
doi:10.1007/3-540-45682-1_29 fatcat:642pifhyvvb6pjkttoxhj6xsti

Optimal Eta Pairing on Supersingular Genus-2 Binary Hyperelliptic Curves [chapter]

Diego F. Aranha, Jean-Luc Beuchat, Jérémie Detrey, Nicolas Estibals
2012 Lecture Notes in Computer Science  
As 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.  ... 
doi:10.1007/978-3-642-27954-6_7 fatcat:bmcheryyl5fgpd3pmjgw476gn4

Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

Reynald Lercier, Christophe Ritzenthaler, Jeroen Sijsling
2015 Mathematics of Computation  
If it vanishes, then the use 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.  ... 
doi:10.1090/mcom3032 fatcat:gbsk56ddofaqbhhutu444wf5di

On explicit descent of marked curves and maps

Jeroen Sijsling, John Voight
2016 Research in Number Theory  
We give 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.  ... 
doi:10.1007/s40993-016-0057-3 fatcat:q4rqwdi46zcuhfn3nmu3bpivdm

Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three

Claus Diem, Emmanuel Thomé
2007 Journal of Cryptology  
We present 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.  ... 
doi:10.1007/s00145-007-9014-6 fatcat:clczgl7xr5b5vgekyvma25fxga

The arithmetic of genus two curves [article]

Lubjana Beshaj, Tony Shaska
2012 arXiv   pre-print
and the arithmetic 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 .  ... 
arXiv:1209.0439v3 fatcat:prlfb6uutjbexc2vl6ajwq7l44
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