A reprogrammable hardware platform is used for the co-design and implementation of a computational intensive mathematical problem, namely the listing of irreducible polynomials over Galois fields of order 3 (GF(3)), The main goal is to accelerate the performance compared to an existing software implementation. This project uses hardware/software co-design methodologies and techniques, and it is completely designed, implemented and evaluated on two distinct platforms, not simply by simulations. FPGAs are used as part of the reconfigurable hardware in both a PCI-based environment and in a more successful System-on-Chip (SOC) platform, which takes advantage of the closely-coupled interconnection between the hardware and software, thus minimizing the communication overhead. The case study, findings and general analysis lead to a possible ideal architecture for future approaches. Moreover, a more general detailed strategy can be seen for the transformation from software to a co-design paradigm, maximizing parallelism.