The complexity bounds of NP-hard problems are one of the main topics in modern cryptography, where the security of a scheme is frequently based on the intractability of the computations. This paper explores the theoretical and practical consequences of NP-hardness in cryptographic designs, such as lattice-based designs, hash functions, and secure multiparty designs. We examine reduction methods in cryptographic proofs to put hardness hypotheses and investigate precise algorithms that have exponential-time complexity characterizing worst-case security parameters. Moreover, the heuristic and approximation methods are compared to the effectiveness of solving real-life cases without to jeopardize the security. Both experimental and analytical results indicate that there are trade-offs between the computational feasibility and cryptographic strength. The results prove that although the precise solutions cannot be offered in large cases, the structured approximations and heuristics can offer the way to implement them that will be efficient, which supports the soundness and utility of NP-hardness in secure system design.