Extracting a uniform random bit-string over Jacobian of Hyperelliptic curves of Genus $2$
This work provides an incremental improvement in cryptography for generating random bits from algebraic curves, relevant to secure cryptographic systems.
The paper tackles the problem of extracting uniformly random bit-strings from the Jacobian of hyperelliptic curves of genus 2, improving upon existing deterministic extractors by reducing the upper bound of the statistical distance using Mumford's representation.
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.