In the new age, most medical digital imaging and media objects (DICOMs) contain images and related medical and demographic information about the patients, which must be kept secret. One of the techniques used to secure medical images is encryption. This study suggests a robust hybrid cryptography algorithm to ensure DICOM privacy. The proposed algorithm uses DNA and RNA encryption algorithms. In the suggested algorithm, in the DNA phase, the DICOM image is encrypted using DNA rules, the XOR operator, and the chaotic map function, and the first cipher image is created. The cipher image is then sent into the RNA phase, where the truth table for RNA and the secret key are used to further encrypt the image. In addition to increasing security for each of the encrypted DICOMs, all keys depend on the patient’s biometric information, such as the patient’s iris or fingerprint. Some of the encryption features such as statistical, differential, and key analysis findings show that the proposed system is resilient and resistant to a variety of attacks.