The technological advances made possible by the Internet, coupled with the unforeseen critical circumstances set in motion by the Covid-19 pandemic, have greatly increased the generation and transmission of medical images every day. Medical image transmission over an unsecured public network threatens the privacy of sensitive patient information. We have, in this paper, designed a new secure color medical image encryption algorithm based on binary plane decomposition, DNA (deoxyribonucleic acid) computing, and the chaotic Rössler dynamical system. At first, a bit-by-bit swap is performed on twenty four binary planes of the input image and encoded using DNA encoding rules. Thereafter, the Rössler system is used to modify the pixel values of the encoded image, which is subsequently decoded. Finally, the ciphered image is obtained by pixel-by-pixel permutation using position sequences. An innovative approach is used to compute keys from the color components of the input image. Extensive performance experiments of the proposed technique is conducted with metrics such as key sensitivity, key space, correlation coefficients (horizontal, diagonal and vertical directions), histograms, information entropy, number of pixel changes rate (NPCR), information entropy, unified average changing intensity (UACI), and encryption time. Comparative analyses have demonstrated that the proposed algorithm is fast, robust and competitive.