Homomorphic Encryption (HE) enables users to securely outsource both the storage and computation of sensitive data to untrusted servers. Not only does HE offer an attractive solution for security in cloud systems, but lattice-based HE systems are also believed to be resistant to attacks by quantum computers. However, current HE implementations suffer from prohibitively high latency. For lattice-based HE to become viable for real-world systems, it is necessary for the key bottlenecks - particularly polynomial multiplication - to be highly efficient. In this paper, we present a characterization of GPU-based implementations of polynomial multiplication. We begin with a survey of modular reduction techniques and analyze several variants of the widely-used Barrett modular reduction algorithm. We then propose a modular reduction variant optimized for 64-bit integer words on the GPU, obtaining a 1.8x speedup over the existing comparable implementations. Next, we explore the following GPU-specific improvements for polynomial multiplication targeted at optimizing latency and throughput: 1) We present a 2D mixed-radix, multi-block implementation of NTT that results in a 1.85x average speedup over the previous state-of-the-art. 2) We explore shared memory optimizations aimed at reducing redundant memory accesses, further improving speedups by 1.2x. 3) Finally, we fuse the Hadamard product with neighboring stages of the NTT, reducing the twiddle factor memory footprint by 50%. By combining our NTT optimizations, we achieve an overall speedup of 123.13x and 2.37x over the previous state-of-the-art CPU and GPU implementations of NTT kernels, respectively.