With the development of cloud computing technology, the security of data in the cloud has become an important issue. Verifying the confidentiality, integrity, and security of data in the cloud has become the main way to ensure the security of data in the cloud. In cryptographic schemes, modular exponential is one of the most basic operations for data verification in the cloud, but modular exponential operations consume more resources due to their energy-consuming calculation. Efficiently and securely outsourcing modular exponentiation to cloud servers becomes a critical challenge. In this paper, we propose a new scheme to address this problem. Specifically, we introduce efficient and secure outsourcing algorithms for modular exponentiation of the form \( u^{x} mod p \) . Our approach is designed for the malicious single-cloud setting, overcoming some limitations in existing methods. For instance, the scheme proposed by Wang et al. is shown to be insecure, while the scheme of Chevalier et al. relies on an unreasonable assumption regarding pre-computation. Furthermore, we demonstrate reusing random pre-computation pairs can lead to more efficient results, reducing computational overhead while maintaining security. These advancements make our proposed scheme both practical and effective for secure outsourcing of modular exponentiation in cloud environments.