The big amount of patients’ information is stored in the different medical organizations. The most part of this data must be stored privately. Therefore, this information must be encrypted. Scientists and engineers are working to develop quantum computers. Google Corporation, Universities Space Research Association and federal agency NASA together with D-WAVE are now ready for the breakthrough of quantum technology. Eventually quantum computers will be capable of cracking the cryptographic codes used today to secure communications, financial transactions and in medical sector. Presently used asymmetric crypto-systems are vulnerable to attacks implemented by quantum computers and the world should have adopted quantum-resistant cryptography. The security of currently used asymmetric systems is based on the problem of calculating discrete logarithms and the large numbers factoring. Some cryptosystems for example RSA – with four thousand bit keys are useful to stand against attacks implemented on classical computer, but are absolutely useless against attacks implemented by quantum computers. To date we meet RSA cryptosystem on almost each step, as it is used by many large organizations, for example, in the hospitals, government institutions, banks, most corporations, government laboratories, universities and in other medical organizations. The fact that the destruction of RSA can entail the rather easy hacking of the most systems, can lead to a real chaos. During the presentation, I will describe and analyze the post-quantum alternatives of RSA. I will analyze their security and efficiency. I will also analyze the existing attacks on these alternatives. The new efficient and secure post-quantum crypto system will also be offered during the talk. I will offer its security and efficiency analysis. The programming implementation of the new post-quantum encryption scheme will also be offered.