Extracting a uniform random bit-string over Jacobian of Hyperelliptic curves of Genus $2$

TL;DR

This paper improves deterministic randomness extractors for hyperelliptic curve Jacobians of genus 2, producing perfectly random bit strings with reduced statistical distance bounds using Mumford's representation.

Contribution

It introduces an enhanced method for extracting uniformly random bits from Jacobians of genus 2 hyperelliptic curves, improving previous extractor bounds.

Findings

Extracts perfectly random bit strings from Jacobians of genus 2 hyperelliptic curves.

Reduces the statistical distance bounds of the extractors.

Uses Mumford's representation for elementary analysis.

Abstract

Here, we proposed an improved version of the deterministic random extractors $SEJ$ and $PEJ$ proposed by R. R. Farashahi in \cite{F} in 2009. By using the Mumford's representation of a reduced divisor $D$ of the Jacobian $J(\mathbb{F}_q)$ of a hyperelliptic curve $\mathcal{H}$ of genus $2$ with odd characteristic, we extract a perfectly random bit string of the sum of abscissas of rational points on $\mathcal{H}$ in the support of $D$. By this new approach, we reduce in an elementary way the upper bound of the statistical distance of the deterministic randomness extractors defined over $\mathbb{F}_q$ where $q=p^n$, for some positive integer $n\geq 1$ and $p$ an odd prime.