Security issues play an important role in almost all modern communication and computer networks. As Internet applications continue to grow dramatically, security requirements have to be strengthened. Hyperelliptic curve cryptosystems (HECC) allow for shorter operands at the same level of security than other public-key cryptosystems, such as RSA or Diffie-Hellman. These shorter operands appear promising for many applications. Hyperelliptic curves are a generalization of elliptic curves and they can also be used for building discrete logarithm public-key schemes. A major part of this work is the development of computer architectures for the different algorithms needed for HECC. The architectures are developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients and underlying finite field. In this work we first summarized the theoretical background of hyperelliptic curve cryptosystems. In order to realize the operation addition and doubling on the Jacobian, we developed architectures for the composition and reduction step. These in turn are based on architectures for arithmetic in the underlying field and for arithmetic in the polynomial ring. The architectures are described in VHDL (VHSIC Hardware Description Language) and the code was functionally verified. Some of the arithmetic modules were also synthesized. We provide estimates for the clock cycle count for a group operation in the Jacobian. The system targeted was HECC of genus four over GF(2^41).