Zero-knowledge proofs (ZKPs) have emerged as a promising solution to address the scalability challenges in modern blockchain systems. This study proposes a methodology for generating and verifying ZKPs to ensure the computational integrity of cryptographic hashing, specifically focusing on the SHA-256 algorithm. By leveraging the Plonky2 framework, which implements the PLONK protocol with FRI commitment scheme, we demonstrate the efficiency and scalability of our approach for both random data and real data blocks from the NEAR blockchain. The experimental results show consistent performance across different data sizes and types, with the time required for proof generation and verification remaining within acceptable limits. The generated circuits and proofs maintain manageable sizes, even for real-world data blocks with a large number of transactions. The proposed methodology contributes to the development of secure and trustworthy blockchain systems, where the integrity of computations can be verified without revealing the underlying data. Further research is needed to assess the applicability of the approach to other cryptographic primitives and to evaluate its performance in more complex real-world scenarios.