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

  • Reza Rezaeian Farashahi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4887)

Abstract

We propose two simple and efficient deterministic extractors for \(J(\mathbb{F}_q)\), the Jacobian of a genus 2 hyperelliptic curve H defined over \(\mathbb{F}_q\), for some odd q. Our first extractor, \(\texttt{SEJ}\), called sum extractor, for a given point D on \(J(\mathbb{F}_q)\), 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, \(\texttt{PEJ}\), called product extractor, for a given point D on the \(J(\mathbb{F}_q)\), outputs the product of abscissas of rational points in the support of D. Provided that the point D is chosen uniformly at random in \(J(\mathbb{F}_q)\), the element extracted from the point D is indistinguishable from a uniformly random variable in \(\mathbb{F}_q\). Thanks to the Kummer surface \(\mathcal{K}\), that is associated to the Jacobian of H over \(\mathbb{F}_q\), we propose the sum and product extractors, \(\texttt{SEK}\) and \(\texttt{PEK}\), for \(\mathcal{K}(\mathbb{F}_q)\). These extractors are the modified versions of the extractors \(\texttt{SEJ}\) and \(\texttt{PEJ}\). Provided a point K is chosen uniformly at random in \(\mathcal{K}\), the element extracted from the point K is statistically close to a uniformly random variable in \(\mathbb{F}_q\).

Keywords

Jacobian Hyperelliptic curve Kummer surface Deterministic extractor 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Reza Rezaeian Farashahi
    • 1
    • 2
  1. 1.Dept. of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB EindhovenThe Netherlands
  2. 2.Dept. of Mathematical Sciences, Isfahan University of Technology, P.O. Box 85145 IsfahanIran

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