In this study, we propose a new cryptographic scheme based on the textbook RSA algorithm associating the concept of continued fractions. The algorithm is designed under the three primary steps of key generation, encryption, and decryption. The key generation of the system is improved to generate a large encryption key without affecting the decryption key. The use of continued fractions gives an additional advantage as it encrypts a string of plaintexts by encrypting only two integers which optimizes the time and the memory consumption. The encryption process and the decryption process are similar to that of the standard RSA algorithm, insomuch the implemented algorithm does not affect the security of the original RSA algorithm which depends on the factoring problem. In addition, as the algorithm uses a large encryption key, it has been proven that the system is secure against the Wiener’s attack.