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

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.