NTTs have proven to be an important calculator in modern public key encryption, particularly in lattice-based and post-quantum cryptography. Mathematical backgrounds and algorithms implementations of NTTs in the context of efficient arithmetic of polynomials over finite field are discussed in this paper. The study addresses the flaws of the conventional convolution algorithms, and suggests simplified Cooley Tukey founded NTT algorithms, which can translate to a lot when it comes to computational complexity. Additionally, it also talks about how NTT can be integrated into the environment of algebraic structures (rings and modules) so as to enable safe and dimensionless encryption. Benchmarking at the experimental level demonstrates the improvements in the computational efficiency, as much as 45 per cent reduction in the processing time as compared to the traditional methods. The results introduce better throughput, reduced latency and capability to run in resource limited environments.