Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations
This work addresses a specific computational bottleneck in cryptographic applications involving hyperelliptic curves, but it is incremental as it builds directly on existing methods.
The paper tackles the problem of computing explicit endomorphisms for divisors with non-disjoint support on the Jacobian of a genus 2 hyperelliptic curve, extending prior work that handled disjoint support cases, and presents new formulae for this scenario and a modified approach for divisor doubling.
We present an efficient endomorphism for the Jacobian of a curve $C$ of genus 2 (hyperelliptic) for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor doubling and addition of divisors with disjoint support in $\mathbb{J}(C)$ using only base field operations. Explicit formulae is presented for this third case and a slightly different approach for divisor doubling.