abstract: Lattice-based Cryptography is an up and coming field of cryptography that utilizes the difficulty of lattice problems to design lattice-based cryptosystems that are resistant to quantum attacks and applicable to Fully Homomorphic Encryption schemes (FHE). In this thesis, the parallelization of the Residue Number System (RNS) and algorithmic efficiency of the Number Theoretic Transform (NTT) are combined to tackle the most significant bottleneck of polynomial ring multiplication with the hardware design of an optimized RNS-based NTT polynomial multiplier. The design utilizes Negative Wrapped Convolution, the NTT, RNS Montgomery reduction with Bajard and Shenoy extensions, and optimized modular 32-bit channel arithmetic for nine RNS channels to accomplish an RNS polynomial multiplication. In addition to a full software implementation of the whole system, a pipelined and optimized RNS-based NTT unit with 4 RNS butterflies is implemented on the Xilinx Artix-7 FPGA(xc7a200tlffg1156-2L) for size and delay estimates. The hardware implementation achieves an operating frequency of 47.043 MHz and utilizes 13239 LUT's, 4010 FF's, and 330 DSP blocks, allowing for multiple simultaneously operating NTT units depending on FGPA size constraints. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2020