G admits a scalar multiplication, [direct sum], possessing the following properties.

with vector addition [direct sum] and scalar multiplication [cross product], such that for all r [member of] R and a, b [member of] G,

Scalar multiplication is the key operation in hyperelliptic curve cryptosystem.

In practical hyperelliptic curve cryptosystems, the vital computation that dominates the whole running time is scalar multiplication, that is, the computation of the repeated divisor adding

2) G admits a

scalar multiplication, [cross product], possessing the following properties.

Then E(K) is closed under addition and positive scalar multiplication.

This proves that [lambda]E(x)=E([lambda]x)[member of]E(K) and hence E(K) is closed under positive scalar multiplication.

The linear structure of L(R) induces addition X + Y and

scalar multiplication [lambda]X, ([lambda] [member of] R) in terms of [alpha] level sets by [[X + Y].

Finally, we prove that

scalar multiplication is continuous.

Additionally, each entity has to compute 6

scalar multiplication and 8 exponentiations.

called

scalar multiplication, which associates with each scalar k in F and a vector a in V a vector k x a in V , called the product of k with a, in such a way that

with vector addition [direct sum] and

scalar multiplication [direct sum], such that for all r [member of] R and a, b [member of] G: