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The Number Theoretic Transform (NTT) enables faster polynomial multiplication and is becoming a fundamental component of next-generation cryptographic systems. NTT hardware designs have two prevalent problems related to design-time flexibility. First, algorithms have different arithmetic structures causing the hardware designs to be manually tuned for each setting. Second, applications have diverse throughput/area needs but the hardware have been designed for a fixed, pre-defined number of processing elements. This paper proposes a parametric NTT hardware generator that takes arithmetic configurations and the number of processing elements as inputs to produce an efficient hardware with the desired parameters and throughput. We illustrate the employment of the proposed design in two applications with different needs: A homomorphically encrypted deep neural network inference (CryptoNets) and a post-quantum digital signature scheme (qTESLA). We propose the first NTT hardware acceleration for both applications on FPGAs. Compared to prior software and high-level synthesis solutions, the results show that our hardware can accelerate NTT up to 28× and 48×, respectively. Therefore, our work paves the way for high-level, automated, and modular design of next-generation cryptographic hardware solutions.