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Group Law Computations on Jacobians of Hyperelliptic Curves [chapter]

Craig Costello, Kristin Lauter
2012 Lecture Notes in Computer Science  
At the same time however, increasing the genus of a hyperelliptic curve significantly increases the computational cost of performing a group operation in the corresponding Jacobian group.  ...  Cantor [6] was the first to give a concrete algorithm for performing computations in Jacobian groups of hyperelliptic curves over fields of odd characteristic.  ...  Acknowledgements We wish to thank Huseyin Hisil and Michael Naehrig for many fixes and improvements to an earlier version of this paper.  ... 
doi:10.1007/978-3-642-28496-0_6 fatcat:gsjxirk34rczvbvwi24c4dy5ce

Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations [article]

Eduardo Ruiz Duarte, Octavio Páez Osuna
2014 arXiv   pre-print
We present an efficient endomorphism for the Jacobian of a curve C of genus 2 (hyperelliptic) for divisors having a Non disjoint support.  ...  This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor doubling and addition of divisors with disjoint support in J(C) using only base field operations.  ...  a geometrical approach for genus 2 hyperelliptic function fields using divisor theory and the Mumford representation of the elements of the Jacobian of the hyperelliptic function field, this approach  ... 
arXiv:1405.5755v1 fatcat:u5kujjmnqnhspeuo2ekulaelh4

Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations

Eduardo Ruiz Duarte, Octavio Páez Osuna
2015 Studia scientiarum mathematicarum Hungarica (Print)  
We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support.  ...  This extends the work of Costello in [12] who calculated explicit formulae for divisor doubling and addition of divisors with disjoint support in J F (C) using only base field operations.  ...  a geometrical approach for genus 2 hyperelliptic function fields using divisor theory and the Mumford representation of the elements of the Jacobian of the hyperelliptic function field, this approach  ... 
doi:10.1556/012.2015.52.2.1313 fatcat:t5tt3fulc5gebmtkdqdvbofuqu

An introduction to hyperelliptic curve arithmetic [chapter]

R. Scheidler
2016 Contemporary Developments in Finite Fields and Applications  
Acknowledgment The author is supported by a Discovery Grant from the National Science and Engineering Research Council (NSERC) of Canada.  ...  Finally, the helpful comments of an anonymous reviewer are appreciated.  ...  To achieve a fixed security level √ q g ≈ 2 n (see [3] for recommended values of n), a genus 1 curve needs to be defined over a field of size q ≈ 2 2n , whereas the field of definition of a genus 2 curve  ... 
doi:10.1142/9789814719261_0019 fatcat:hiiwblf5sbg2tbrta44vqxcxem

Comparative Study of Hyperelliptic Curve Cryptosystem over Prime Field and Its Survey

P. Vijayakumar, V. Vijayalakshmi, G. Zayaraz
2014 International Journal of Hybrid Information Technology  
Jacobians of Hyperelliptic curves of genus 2 or 3 require less processing time for divisor generation, key generation, encryption and decryption than genus 4, 5 and 6.  ...  This paper shows the performance of HECC over genus curve 2, 3, 4, 5, and 6 for length of prime field F q = 45 and 65.  ...  Polynomial chosen for genus 2, 3, 4, 5 and 6 over prime field F p is given below Genus g=2 (4) Genus g=3 (5) Genus g=4 (6) Genus g=5 (7) Genus g=6 (8) Jacobian of Hyperelliptic Curve The Jacobian of  ... 
doi:10.14257/ijhit.2014.7.1.11 fatcat:4fousv5vbvddnl5wx4ktdkwewm

Cryptographic aspects of real hyperelliptic curves

Michael J. Jacobson, Renate Scheidler, Andreas Stein
2010 Tatra Mountains Mathematical Publications  
We then describe recent improvements to infrastructure arithmetic, including explicit formulas for divisor arithmetic in genus 2, and advances in solving the infrastructure discrete logarithm problem,  ...  We review previously proposed cryptographic protocols and discuss the infrastructure of a real hyperelliptic curve, the mathematical structure underlying all these protocols.  ...  The authors are grateful to an anonymous referee's helpful suggestions for improvement of this article.  ... 
doi:10.2478/v10127-010-0030-9 fatcat:2t7ft7jaizd2tkxmx3kwydnghq

A remark on a Nagell-Lutz type statement for the Jacobian of a curve of genus 2 and a (2, 3, 6) quasi-torus decomposition of a sextic with 9 cusps [article]

Hiro-o Tokunaga, Yukihiro Uchida
2018 arXiv   pre-print
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over C(t).  ...  As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.  ...  Mumford representations derive from the construction of the Jacobian variety of a hyperelliptic curve given by Mumford [13] . Let d = i m i P i be an effective affine semi-reduced divisor.  ... 
arXiv:1808.10187v2 fatcat:sawincwgwrf73gv3hvudxdvkiq

Performance Study of genus 3 Hyperelliptic Curve Cryptosystem

Daya Gupta, Asok De, Kakali Chatterjee
2012 Journal of Information Processing Systems  
This paper explores various possible attacks to the discrete logarithm in the Jacobian of a Hyperelliptic Curve (HEC) and addition and doubling of the divisor using explicit formula to speed up the scalar  ...  We can construct genus 3 HECC on 54-bit finite fields in order to achieve the same security level as 160-bit ECC or 1024-bit RSA due to the algebraic structure of Hyperelliptic Curve.  ...  They found a sub-exponent time algorithm to solve the DL in the Jacobian of hyperelliptic curves of a big genus over a finite field.  ... 
doi:10.3745/jips.2012.8.1.145 fatcat:rzrst2vsrjfrpjjp5s6soeopji

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 also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework.  ...  The work of the first author has been supported by a National Science Foundation Graduate Research Fellowship.  ... 
arXiv:0908.3731v2 fatcat:6nnfdtdi2rgnzpeyk642nf3mum

Fast Arithmetic on Jacobians of Picard Curves [chapter]

Stéphane Flon, Roger Oyono
2004 Lecture Notes in Computer Science  
In this paper we present a fast addition algorithm in the Jacobian of a Picard curve over a finite field Fq of characteristic different from 3.  ...  In this article, we find explicit formulae for computing in the Jacobian of a Picard curve, basing us on some geometric aspects of these curves.  ...  We also thank the referees for their comments, and Roberto Avanzi for pointing out to us the use of fast endomorphisms.  ... 
doi:10.1007/978-3-540-24632-9_5 fatcat:ger33m5xabblbai5pak3bg27ma

Efficient Hyperelliptic Arithmetic Using Balanced Representation for Divisors [chapter]

Steven D. Galbraith, Michael Harrison, David J. Mireles Morales
2008 Lecture Notes in Computer Science  
We discuss arithmetic in the Jacobian of a hyperelliptic curve C of genus g.  ...  The traditional approach is to fix a point P∞ ∈ C and represent divisor classes in the form E − d(P∞) where E is effective and 0 ≤ d ≤ g.  ...  The third author thanks CONACyT for its financial support.  ... 
doi:10.1007/978-3-540-79456-1_23 fatcat:sa6jkdyfonh3pl5w3a3ntvfita

Fast Jacobian arithmetic for hyperelliptic curves of genus 3

Andrew Sutherland
2019 The Open Book Series  
We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2.  ...  Here we address the general case, in which we do not assume the existence of a rational Weierstrass point, using a balanced divisor approach.  ...  The explicit formulas presented on those pages were typeset using latex source generated by an automated script that reads an executable version of verified source code; they should thus be free of the  ... 
doi:10.2140/obs.2019.2.425 fatcat:qzpotrbthjdkjbpky5rreq6iai

A double large prime variation for small genus hyperelliptic index calculus

P. Gaudry, E. Thomé, N. Thériault, C. Diem
2007 Mathematics of Computation  
Computer experiments show that for hyperelliptic curves of genus three, our first algorithm surpasses Pollard's Rho method even for rather small field sizes.  ...  In this article, we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation.  ...  Acknowledgments The first three authors thank Antoine Lejay who helped with the probabilistic statements that occurred in the first proof of Theorem 1.  ... 
doi:10.1090/s0025-5718-06-01900-4 fatcat:qfqbxycaorfi3jwg6uwgh42uk4

Group Law Algorithms for Jacobian Varieties of Curves over Finite Fields

Ran Cohen
2008 Algebraic Geometry and Its Applications  
This paper reviews the group law algorithms of nonsingular C ab curves, and provides implementation results comparing it to an algorithm for computing in the generalized Jacobian of C A curves.  ...  Acknowledgment The results of this paper were obtained during my Master studies at Bar-Ilan University.  ...  I would like to express deep gratitude to my supervisor Professor Boris Kunyavskii whose guidance and support were crucial for the successful completion of this project.  ... 
doi:10.1142/9789812793430_0011 fatcat:gre7vztdfrb2zj5l5m6vupd4si

Page 497 of Mathematical Reviews Vol. 49, Issue 2 [page]

1975 Mathematical Reviews  
Let M, be the moduli space of curves of genus g and let M, be its compactification in the sense of P. Deligne and D. Mumford [Inst. Hautes Etudes Sci. Publ. Math.  ...  The period mapping 7:M,—-S, extends (Theorem 3) naturally to a mapping 7: M,—S, (intuitively 7 associates with a stable curve the Jacobian of its normalization).  ... 
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