Cloud computing enables users to process and store data remotely on high-performance computers and servers by sharing data over the Internet. However, transferring data to clouds causes unavoidable privacy concerns. Here, we present a synthesis framework to design coding mechanisms that allow sharing and processing data in a privacy-preserving manner without sacrificing data utility and algorithmic performance. We consider the setup where the user aims to run an algorithm in the cloud using private data. The cloud then returns some data utility back to the user (utility refers to the service that the algorithm provides, e.g., classification, prediction, AI models, etc.). To avoid privacy concerns, the proposed scheme provides tools to co-design: 1) coding mechanisms to distort the original data and guarantee a prescribed differential privacy level; 2) an equivalent-but-different algorithm (referred here to as the target algorithm) that runs on distorted data and produces distorted utility; and 3) a decoding function that extracts the true utility from the distorted one with a negligible error. Then, instead of sharing the original data and algorithm with the cloud, only the distorted data and target algorithm are disclosed, thereby avoiding privacy concerns. The proposed scheme is built on the synergy of differential privacy and system immersion tools from control theory. The key underlying idea is to design a higher-dimensional target algorithm that embeds all trajectories of the original algorithm and works on randomly encoded data to produce randomly encoded utility. We show that the proposed scheme can be designed to offer any level of differential privacy without degrading the algorithm's utility. We present two use cases to illustrate the performance of the developed tools: privacy in optimization/learning algorithms and a nonlinear networked control system.