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The authority of user personal health records (PHRs) is usually determined by the owner of a cloud computing system. When a PHR file is accessed, a dynamic access control algorithm must be used to authenticate the users. The proposed dynamic access control algorithm is based on a novel Lagrange interpolation polynomial with timestamps, mainly functioning to authenticate the users with key information. Moreover, the inclusion of timestamps allows user access within an approved time slot to enhance the security of the healthcare cloud system. According to the security analysis results, this healthcare cloud system can effectively resist common attacks, including external attacks, internal attacks, collaborative attacks and equation-based attacks. Furthermore, the overall computational complexity of establishing and updating the polynomials is O(<italic>n</italic>*<italic>m</italic>* (log <italic>m</italic>)²), which is a promising result, where <italic>m</italic> denotes the degree of $ polynomial~G\left(x, y\right) $ and <italic>n</italic> denotes the number of secure users in the hierarchy.