Extractors for Jacobian of Hyperelliptic Curves of Genus 2 in Odd Characteristic

@inproceedings{Farashahi2007ExtractorsFJ,
  title={Extractors for Jacobian of Hyperelliptic Curves of Genus 2 in Odd Characteristic},
  author={Reza Rezaeian Farashahi},
  booktitle={IMACC},
  year={2007}
}
We propose two simple and efficient deterministic extractors for J(Fq), the Jacobian of a genus 2 hyperelliptic curve H defined over Fq, for some odd q. Our first extractor, SEJ, called sum extractor, for a given point D on J(Fq), outputs the sum of abscissas of rational points on H in the support of D, considering D as a reduced divisor. Similarly the second extractor, PEJ, called product extractor, for a given point D on the J(Fq), outputs the product of abscissas of rational points in the… 
Extracting a uniform random bit-string over Jacobian of Hyperelliptic curves of Genus 2
TLDR
By using the Mumford’s representation of a reduced divisor D of the Jacobian J(Fq) of a hyperelliptic curve H of genus 2 with odd characteristic, a perfectly random bit string is extracted of the sum of abscissas of rational points on H in the support of D.
Extractors for Jacobians of Binary Genus-2 Hyperelliptic Curves
TLDR
It is shown that, if D in J(\mathbb{F}_q) is chosen uniformly at random, the bits extracted from Dare are indistinguishable from a uniformly random bit-string of length n.
Linear complexity of some sequences derived from hyperelliptic curves of genus 2
TLDR
The hyperelliptic analogue of the congruential generator defined by W n = W n − 1 + D for n ≥‬1 and D, W 0 ∈ J C is considered.
Kummer surfaces for primality testing
TLDR
An algorithm is provided capable of proving the primality or compositeness of most of the integers in these families and the necessary steps to implement this algorithm in a computer are discussed.
Pseudorandom Bits From Points on Elliptic Curves
TLDR
These bounds confirm several recent conjectures of Jao, Jetchev, and Venkatesan, related to extracting random bits from various sequences of points on the elliptic curves.
Curves and Jacobians : number extractors and efficient arithmetic
TLDR
The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.

References

SHOWING 1-10 OF 29 REFERENCES
Formulae for Arithmetic on Genus 2 Hyperelliptic Curves
  • T. Lange
  • Mathematics, Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2004
TLDR
This article presents explicit formulae to perform the group operations for genus 2 curves and introduces a new system of coordinates and state algorithms showing that doublings are comparably cheap and no inversions are needed.
Extractors for binary elliptic curves
TLDR
It is shown that if a point P is chosen uniformly at random in G, the bits extracted from the point P are indistinguishable from a uniformly random bit-string of length ℓ.
Aspects of Hyperelliptic Curves over Large Prime Fields in Software Implementations
  • R. Avanzi
  • Mathematics, Computer Science
    CHES
  • 2004
TLDR
An ad-hoc arithmetic library is developed, designed to remove most of the overheads that penalize implementations of curve-based cryptography over prime fields, which results in a performance much closer to the performance of elliptic curves than previously thought.
Review of "Handbook of Elliptic and Hyperelliptic Curve Cryptography by H. Cohen and G. Frey", Chapman & Hall/CRC, 2006, 1-58488-518-1
1 Overview Elliptic curve cryptography was introduced in the mid 1980s and is now finding applicability in many public key situations. In particular, it provides a level of security comparable to
An Algorithm for Solving the Discrete Log Problem on Hyperelliptic Curves
  • P. Gaudry
  • Computer Science, Mathematics
    EUROCRYPT
  • 2000
TLDR
An index-calculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields and the breaking of a cryptosystem based on a curve of genus 6 recently proposed by Koblitz is described.
Montgomery Scalar Multiplication for Genus 2 Curves
TLDR
This work generalizes the Montgomery method for scalar multiplication to the jacobian of genus 2 curves to obtain an algorithm that is competitive compared to the usual methods of scalarmultiplication and that has additional properties such as resistance to timings attacks.
Fast genus 2 arithmetic based on Theta functions
TLDR
This work derives fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Montgomery ladder, which can be used to design very efficient genus 2 cryptosystems that should be faster than elliptic curve cryptosSystems in some hardware configurations.
Montgomery Addition for Genus Two Curves
  • T. Lange
  • Computer Science, Mathematics
    ANTS
  • 2004
TLDR
This work has shown that the curve arithmetic proposed by Montgomery requires a comparably small number of field operations to perform a scalar multiplication but at the same time achieves security against non-differential side channel attacks.
Computing in the Jacobian of a hyperelliptic curve
TLDR
A reduction algorithm is presented which is asymptotically faster than that of Gauss when the genus g is very large and the Jacobian of a hyperelliptic curve is studied.
Extracting bits from coordinates of a point of an elliptic curve
  • Nicolas Gürel
  • Mathematics, Computer Science
    IACR Cryptol. ePrint Arch.
  • 2005
TLDR
This paper presents a new deterministic method to extract bits from KAB when G is an elliptic curve defined over a quadratic extension of a finite field.
...
...