The thesis concerns properties of Jacobians of genus two curves defined over a finite field. Such Jacobians have a wide range of applications in data security; e.g. netbanking and digital signature.

New properties of the Jacobians are proved; here, a description of the embedding of $\ell$-torsion points on the Jacobian and the matrix representation of the Frobenius endomorphism are the central results.

The properties of the Jacobians are exploited to develop a procedure to find generators of the $\ell$-torsion points on the Jacobian.