The adoption of cloud computing solutions is an established reality in government agencies and in small, medium, and large companies due to procurement easiness and the variety of available services, as well as its low cost compared to the acquisition and management of own infrastructures. Among the most used services is cloud file storage, and the security of this storage has been an essential subject of recent research, particularly customer data integrity. Thus, this article proposes a solution for the monitoring of the integrity of files stored in the cloud, based on the use of smart contracts in Blockchain Networks, symmetric encryption, and computational trust. The proposed solution consists of a protocol that provides confidentiality, decentralization, audit availability, and the secure sharing of file integrity monitoring results, without overloading the services involved, as well as an unabridged reference implementation which was used to validate the proposal. The results obtained during the validation tests have shown that the solution is feasible and faultless in detecting corrupted files. These tests also confirmed that the sharing of integrity monitoring results, coupled with the application of computational trust techniques, significantly increased the efficiency of the proposed solution.