Periods of hyperelliptic integrals expressed in terms of θ -constants by means of Thomae formulae

2007 ◽  
Vol 366 (1867) ◽  
pp. 1005-1024 ◽  
Author(s):  
Victor Enolski ◽  
Peter Richter
Keyword(s):  
Hyperelliptic Curves ◽  
Genus 2

Expressions for the periods of first- and second-kind integrals on hyperelliptic curves are given in terms of θ -constants. They are derived with the help of Thomae's classical formulae and Picard–Fuchs equations for complete integrals as functions of the parameters of the curves. The example of genus 2 is considered in detail.

scholarly journals Cryptographic aspects of real hyperelliptic curves

2010 ◽  
Vol 47 (1) ◽  
pp. 31-65 ◽  
Author(s):  
Michael J. Jacobson ◽  
Renate Scheidler ◽  
Andreas Stein
Keyword(s):  
Hyperelliptic Curve ◽  
Discrete Logarithm ◽  
Mathematical Structure ◽  
Hyperelliptic Curves ◽  
Cryptographic Protocols ◽  
Discrete Logarithm Problem ◽  
Explicit Formulas ◽  
Genus 2

Abstract In this paper, we give an overview of cryptographic applications using real hyperelliptic curves. We review previously proposed cryptographic protocols and discuss the infrastructure of a real hyperelliptic curve, the mathematical structure underlying all these protocols. 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, whose presumed intractability is the basis of security for the related cryptographic protocols.

scholarly journals Explicit formulas for real hyperelliptic curves of genus 2 in affine representation

2011 ◽  
Vol 5 (4) ◽  
pp. 623-666 ◽  
Author(s):  
Stefan Erickson ◽  
Michael Jacobson, Jr. ◽  
Andreas Stein
Keyword(s):  
Hyperelliptic Curves ◽  
Explicit Formulas ◽  
Genus 2

Supersingular hyperelliptic curves of genus 2 over finite fields

2005 ◽  
Vol 163 (2) ◽  
pp. 565-576 ◽  
Author(s):  
Young Ju Choie ◽  
Eun Kyung Jeong ◽  
Eun Jeong Lee
Keyword(s):  
Finite Fields ◽  
Hyperelliptic Curves ◽  
Genus 2

On Indifferentiable Hashing into the Jacobian of Hyperelliptic Curves of Genus 2

2017 ◽  
pp. 205-222 ◽  
Author(s):  
Michel Seck ◽  
Hortense Boudjou ◽  
Nafissatou Diarra ◽  
Ahmed Youssef Ould Cheikh Khlil
Keyword(s):  
Hyperelliptic Curves ◽  
Genus 2
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