A cloud computing platform delivers a cost-efficient path for cloud users to store and access data privately over remote storage via an Internet connection. Medical data is stored in the cloud since it is scalable, secure, reliable, provides ubiquitous access, and is highly available. It is required to be isolated in terms of physical, network, and operational for protecting data from internal threats or external cyber-attacks. It can be achievable through hybrid cryptography in which symmetric, as well as asymmetric cryptosystem, is being utilized. To assure the confidentiality and integrity of medical data, an improved Robust S-box-based Advanced Encryption Standard (IRS-AES) is proposed with Runge-Kutta Optimization (RKO) algorithm. RKO generates an enhanced and secured RS-box through the computation of the Mackey-Glass equation. Initially, the medical data are compressed using Improved Huffman Coding (IHC). The Deoxyribonucleic Acid (DNA)-based Modified Elliptic Curve Cryptography (MECC) algorithm is introduced for key generation, and the best key is selected with Bald Eagle Search optimization (BES) algorithm. Finally, the medical data are encrypted IRS-AES algorithm and stored in the cloud. The proposed RS-box security strength is evaluated using non-linearity, Strict Avalanche Criterion (SAC), Differential Probability (DP), Bit Independence Criterion (BIC), and Linear Probability (LP) parameters. The efficiency of the proposed algorithm is evaluated with image, audio, and video dataset. The evaluation metrics such as communication overhead, file upload time, computational cost, Mean Square Error (MSE), encryption and decryption time, Peak Signal-to-Noise Ratio (PSNR), key generation time, and Signal-to-Noise Ratio (SNR) are used for validating the proposed approach. The performance of the proposed approach is enhanced with the communication overhead of 11.51 for the image dataset. MSE and PSNR obtained are 16.2 and 8.465 for an audio dataset, 75.21 and 3.5 for the video dataset.