Presented herein is a novel algorithm for multi-round, zero-knowledge proof (ZKP), devised specifically for authenticating factorisation proofs within a variety of cryptographic applications. This advanced algorithm, while maintaining computational complexity within acceptable bounds, offers a secure and proficient solution. The functionality of the algorithm is marked by multiple rounds of interaction between the Prover and Verifier. Initially, the Prover generates a random value and calculates a commitment. Subsequently, the Verifier issues a random challenge, eliciting a computed response from the Prover. To validate the proof, the Verifier verifies the equality of the commitment and the computed response. Efficaciousness of the proposed multi-round ZKP algorithm is demonstrated across diverse input sizes and parameters. Results indicate a success rate exceeding 90% on average, showcasing the robustness of the method. The recurring interaction between the Verifier and Prover enhances the Prover's authentication, thereby improving the algorithm’s reliability. Implementation of the algorithm, achievable through standard cryptographic tools and protocols, can fortify the security of multiple cryptographic applications. A significant application can be found in Digital Identity Management Systems (DIMS). Currently, these systems are vulnerable to a myriad of threats, including identity spoofing, data breaches, and internal security risks. The application of the ZKP algorithm can simultaneously augment security and withhold sensitive information, potentially transforming the DIMS security landscape. Future research may focus on improving the efficiency and scalability of the multi-round ZKP algorithm. There also remains a vast potential for exploring additional applications of this technique within various cryptographic domains.