According to the National Institute of Standard and Technology (NIST), the security level of RSA is safe when it is N-bit modulus ≥2048 bits. Because of this, the processing time to generate asymmetric keys also increases. Taking this into account, an efficient and non-shareable Public Key Exponent Secure Scheme (ENPKESS) is proposed by using a non-linear Diophantine equation to have high security against side-channel attacks like timing attacks. This scheme has three-stage of encryption and two-stage of decryption, whereas other schemes like ESR and RSA has one level in encryption and decryption. Due to this, extraction of the secret key is extremely hard to determine from our public exponents α,R,N. Our methodology is well suited for secure cloud computing environments used in the Internet of Things (IoT). Here we have also applied the Knapsack method to encrypt our ENPKESS keys to enrich high security in cloud systems. We show a strong performance evaluation on standard RSA, Enhanced and Secured RSA Key Generation Scheme (ESRKGS), and ENPKESS on its key generation, encryption and decryption by varying the N-bit moduli size up to 10K bits. From the overall result, ENPKESS consumes 89% of standard RSA and 27% of ESRKGS. © 2019 Elsevier B.V.