The concept of entropy has been introduced in statistical methods to measure the amount of information contained in a random observation, and it plays a crucial role in various fields, especially in data security. This paper focuses on the semi-parametric shrinkage estimation of Shannon entropy and past entropy measures of the geometric distribution under complete, right, and time-censored sampling procedures. Shannon entropy, a key measure of uncertainty, along with past entropy (or min-entropy), which assesses the least predictable outcomes, plays a crucial role in ensuring strong data security, particularly in cryptographic systems and secure communications. While most existing literature addresses estimating these entropy measures for continuous distributions, this paper evaluates shrinkage estimators to enhance the efficiency of the ordinary semi-parametric least squares estimator for geometric distributions. This study explores the constant shrinkage factor and modified Thomson-type estimators, evaluating their effectiveness against traditional methods such as maximum likelihood estimators. Empirical investigations conducted with simulated samples indicate that shrinkage estimators consistently outperform maximum likelihood estimators, showcasing better relative efficiency. These results emphasize the potential of shrinkage estimators to enhance entropy-based measures in data security applications, which can lead to more robust cryptographic key generation, password strength analysis, and intrusion detection.