Efficient explicit formulae for genus 2 hyperelliptic curves over prime fields and their implementations

We analyze all the cases and propose the corresponding explicit formulae for computing 2D(1) + D-2 in one step from given divisor classes D-1 and D-2 on genus 2 hyperelliptic curves defined over prime fields. Compared with naive method, the improved formula can save two field multiplications and one field squaring each time when the arithmetic is performed in the most frequent case. Furthermore, we present a variant which trades one field inversion for fourteen field multiplications and two field squarings by using Montgomery's trick to combine the two inversions. Experimental results show that our algorithms can save up to 13% of the time to perform a scalar multiplication on a general genus 2 hyperelliptic curve over a prime field, when compared with the best known general methods.