RSA was developed in 1978 and it has been used widely. The Random Insertion Method (RIM) in cryptography is a newly devised technique incorporating randomness into the encryption process. In this work, we find new transcendental numbers that are useful for encryption algorithms. Combining the concepts of RSA and RIM, three types of encryption and decryption procedures are developed. Computational complexity required in decrypting the encryption by the proposed cryptosystem in terms of the size of the encryption is analysed. It demonstrates that compared to RSA using a public key, the number of bit operations \(\mathscr {T}_{RSA-RIM}\) needed for decryption is significantly higher than that of traditional RSA \(\mathscr {T}_{RSA}\) : \(\mathscr {T}_{RSA} \ll \mathscr {T}_{RSA-RIM}\) . Since the encryption of a message in our joint RSA-RIM cryptosystem makes use of both the RSA and RIM, the resulting encryption has high randomness. It has the needed confusion and diffusion to the interceptor; further, it is not necessary to make blocks of the message even though the original message is very long. For the message encryption, RSA uses the power of the numerical message which comes from the numerical value of the alphabets, whereas our approach uses numerical properties independent of alphabet values. This gives a very secure encryption of messages even if intercepted.