As the cloud computing and mass data sharing develop, data integrity and privacy has become an imperative issue. Conventional remote data auditing techniques tend to reveal sensitive data or they have high computational cost. In order to overcome these shortcomings, the Fully Homomorphic Encryption enhanced Remote Method Invocation (FHEbRMI) mechanism that includes a combination of the Modified Least Squares (MLS) optimization model and the proposed cloud auditing security and efficiency are proposed in this paper. The suggested system provides an encrypted data auditing system, which involves RMI-based communication, to enable the client, server, and third-party auditor to perform their verification functions remotely without the disclosure of the plaintext data. An actual execution of the suggested structure is introduced, such as secure key generation, trapdoor-based dimensionality reduction, ciphertext multiplication, and optimized homomorphic functions. Moreover, the RMI interface provides a smooth communication among the distributed nodes and increases the scalability and minimizes transmission delays. A comparative study with the recent homomorphic-based auditing schemes like blockchain-assisted, certificateless and lattice-based FHE model reveals that the proposed FHEbRMI-MLS model has better performance in terms of encryption/decryption latency, computational cost, and encryption overhead. The experimental performance is indicative of an average 37 and 42 factor in speed of encryption and enhancement of computational efficiency respectively with respect to the traditional FHE models. This paper presents a viable, privacy-friendly auditing framework of clouds which guarantees the end-to-end encrypted verification without sacrificing the efficiency.