Lossless data compression algorithms are widely used by data communication systems and data storage systems to reduce the amount of data transferred and stored. GZIP is a popular, patent-free compression program that delivers good compression ratios. This paper presents hardware implementations for the LZ77 encoders and Huffman encoders that form the basis for a full hardware implementation of a GZIP encoder. The designs have been implemented as state machines in VHDL in such a way that they are suitable for implementation using either FPGA or ASIC technologies. Performance metrics and resource utilization results obtained for a prototype implementation running on an Altera DE2 board are presented. Ultimately, the goal is to utilized the LZ77 encoders and Huffman encoders described in this paper to build a fully-functional, hardware design for a GZIP encoder that could be used in data communication systems and data storage systems to boost overall system performance.