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Hyperelliptic Curves with Compact Parameters

Ezra Brown, Bruce T. Myers, Jerome A. Solinas
2005 Designs, Codes and Cryptography  
We present a family of hyperelliptic curves whose Jacobians are suitable for cryptographic use, and whose parameters can be specified in a highly efficient way.  ...  This is done via complex multiplication and identity-based parameters. We also present some novel computational shortcuts for these families.  ...  Introduction In our earlier paper [1] we introduced the notion of elliptic curves with compact parameters.  ... 
doi:10.1007/s10623-004-1718-0 fatcat:kvmx3ihpg5e5vbrlx4clt7bnki

Compact Representation of Domain Parameters of Hyperelliptic Curve Cryptosystems [chapter]

Fangguo Zhang, Shengli Liu, Kwangjo Kim
2002 Lecture Notes in Computer Science  
We discussed how to represent the domain parameters of HCC in a compact way.  ...  To achieve the same level of security, hyperelliptic curve cryptosystems (HCC) use a smaller field than elliptic curve cryptosystems (ECC).  ...  How to eradicate an parameter in the equation of an hyperelliptic curve is also discussed. We also give the number of bits to represent the order of the Jacobian.  ... 
doi:10.1007/3-540-45450-0_17 fatcat:ybgewadea5hlbgkzuonveijhue

Period Constraints on Hyperelliptic Branch Points [article]

James S. Wolper
2021 arXiv   pre-print
contain any period matrices of hyperelliptic curves.  ...  Information Theoretic analysis of the periods of a hyperelliptic curve provides more information about the well--known but abstract relationship between the branch points and the periods.  ...  In one direction, suppose that Alice wants to tell Bob about a compact curve (ie, Riemann Surface) with genus g ≥ 2.  ... 
arXiv:2106.15696v1 fatcat:ko6dwfpxjnem5pymxvgkn2gzau

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

Ernesto Girondo, Gabino González-Diez
2003 Transactions of the American Mathematical Society  
differential equation with singularities at the values a i .  ...  Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries.  ...  The family of curves y 2 = (x n − 1)(x n − λ) In this section we study the 1-parameter family of hyperelliptic curves of genus n − 1 C λ,n = y 2 = (x n − 1)(x n − λ) , λ ∈ R \ {0, 1}, n ≥ 3, whose branching  ... 
doi:10.1090/s0002-9947-03-03441-x fatcat:3hzymutglrdovej7ppaeh3fh2e

Hyperelliptic solutions of KdV and KP equations: re-evaluation of Baker's study on hyperelliptic sigma functions

Shigeki Matsutani
2001 Journal of Physics A: Mathematical and General  
Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations were constructed for a given curve y^2 = f(x) whose genus is three.  ...  Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator D, identities of Pfaffians, symmetric functions, hyperelliptic σ-function and -functions; _μν = -∂_μ∂_νσ  ...  Although we will mainly deal with a curve with genus three, we will give definitions and expressions of hyperelliptic curves with general genus for later convenience. Notation 1.  ... 
doi:10.1088/0305-4470/34/22/312 fatcat:w7w5wrpln5cxzn2iut7wfq3cxa

On the structure of the Whittaker sublocus of the moduli space of algebraic curves

Ernesto Girondo, Gabino González-Diez
2006 Proceedings of the Royal Society of Edinburgh. Section A Mathematics  
This is the subset of points representing hyperelliptic curves which satisfy Whittaker's conjecture on the uniformization of hyperelliptic curves via monodromy of Fuchsian differential equations.  ...  MS received ; ) We prove the compactness of Whittaker sublocus of moduli space of Riemann surfaces (complex algebraic curves).  ...  sphereĈ = C∪{∞}, as a hyperelliptic algebraic curve.  ... 
doi:10.1017/s0308210500004583 fatcat:waigunffprdnrddl3y254ofdwi

Page 8704 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
++,A2¢42 are branch points on the hyperelliptic curve &, and W% is a g xg matrix of a-periods of nonnormalized holomorphic differentials on this curve.  ...  Summary: “A compact Riemann surface X of genus g is called an (M — 1)-surface if it admits an anticonformal involution that fixes g simple closed curves, the second maximum number by Harnack’s theorem.  ... 

Page 74 of Mathematical Reviews Vol. 51, Issue 1 [page]

1976 Mathematical Reviews  
3 or |L|=|2B+I'|, where B is an irreducible curve of genus 2 and I a rational curve with B['=1.  ...  24 then either (i) F contains an irreducible curve E with E-L=2 and arithmetic genus | or (ii) F contains an irreducible curve B with |L| =|2B| and arithmetic genus 2.  ... 

Page 2514 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
Let M be an elliptic-hyperelliptic compact Riemann surface of genus at least 4.  ...  The problem studied in this paper is the construction of meromor- phic homomorphisms between flat vector bundles over compact Riemann surfaces with prescribed zeros and poles (with multiplic- ities).  ... 

Construction of Hyperbolic Signal Sets from the Uniformization of Hyperelliptic Curves [article]

Erika Patricia Dantas de Oliveira Guazzi, Reginaldo Palazzo Junior
2020 arXiv   pre-print
of the tessellation p,q and the degree of the hyperelliptic curve is established.  ...  In this paper, we present a new approach to the problem of designing hyperbolic signal sets matched to groups by use of Whittaker's proposal in the uniformization of hyperelliptic curves via Fuchsian differential  ...  of the Fuchsian subgroup associated with the fundamental region which uniformizes the given hyperelliptic curve; • To establish a relation between the degree of the hyperelliptic curve and the parameters  ... 
arXiv:2009.05563v1 fatcat:dms6rybbarb6rekg2nen4x4jeu

The Deligne–Mumford and the Incidence Variety Compactifications of the Strata of \Omega \protect \mathcal{M}_{g}

Quentin Gendron
2018 Annales de l'Institut Fourier  
of compact type.  ...  We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three.  ...  In fact, since hyperelliptic curves are covers of degree two above a curve of compact type, many ideas will work in this case. Proof.  ... 
doi:10.5802/aif.3187 fatcat:anxlj7wnfvaircgvqmu2zcvqua

Introduction to Compact Riemann Surfaces [chapter]

Alexander I. Bobenko
2010 Lecture notes in mathematics  
Hyperelliptic Riemann surfaces Let R be a compact Riemann surface of a hyperelliptic curve as in Theorem 1.  ...  The Riemann surface C can be made a compact Riemann surfacê Hyperelliptic curves.  ... 
doi:10.1007/978-3-642-17413-1_1 fatcat:oqdmbeqfkbfljk3mxos2qvlaku

Page 191 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
Finally, he deals with equations of hyperelliptic surfaces of Belyi type.  ...  Fi- nally, the correct reference concerning the classification of groups of automorphisms of hyperelliptic Riemann surfaces is not the re- viewer’s monograph [Groups of automorphisms of compact Riemann  ... 

A Connection Between Geometry and Dynamical Systems

A. Lesfari, A. Elachab
2005 Journal of Dynamical Systems and Geometric Theories  
curve of genus 2. e) We show that at some special values of the parameters λ 1 and λ 2 , we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries  ...  This paper deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the anisotropic harmonic oscillator in a radial quartic potential.  ...  Since Γ is hyperelliptic of genus 2, it has 6 distinct Weierstrass points (Indeed, suppose C is a smooth hyperelliptic curve of genus g ≥ 2 with : C → P 1 (C) the two-sheeted map, then all the branch points  ... 
doi:10.1080/1726037x.2005.10698486 fatcat:6gegentdkjdhvdlx2ab7vcut4u

Information Theory and Moduli of Riemann Surfaces [article]

James S. Wolper
2013 arXiv   pre-print
The results here show that, with high probability, any set of 3g-3 periods form moduli for the surface.  ...  One interpretation of Torelli's Theorem, which asserts that a compact Riemann Surface X of genus g > 1 is determined by the g(g+1)/2 entries of the period matrix, is that the period matrix is a message  ...  Now suppose that Alyce wants to describe a compact Riemann surface with genus g > 1 to Bob.  ... 
arXiv:1309.2304v1 fatcat:hdzhrzrionecdo4ijkx7ejwt4i
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