Recommendation systems powered by artificial intelligence (AI) are widely used to improve user experience. However, AI inevitably raises privacy leakage and other security issues due to the utilization of extensive user data. Addressing these challenges can protect users' personal information, benefit service providers, and foster service ecosystems. Presently, numerous techniques based on differential privacy have been proposed to solve this problem. However, existing solutions encounter issues such as inadequate data utilization and a tenuous trade-off between privacy protection and recommendation effectiveness. To enhance recommendation accuracy and protect users' private data, we propose ID-SR, a novel privacy-preserving social recommendation scheme for trustworthy AI based on the infinite divisibility of Laplace distribution. We first introduce a novel recommendation method adopted in ID-SR, which is established based on matrix factorization with a newly designed social regularization term for improving recommendation effectiveness. We then propose a differential privacy-preserving scheme tailored to the above method that leverages the Laplace distribution's characteristics to safeguard user data. Theoretical analysis and experimentation evaluation on two publicly available datasets demonstrate that our scheme achieves a superior balance between privacy protection and recommendation effectiveness, ultimately delivering an enhanced user experience.