The Secure Sockets Layer (SSL) protocol was developed by Netscape Communications to provide application-independent secure communication over the Internet for protocols such as the Hypertext Transfer Protocol (HTTP). The SSL protocol is intended to provide a practical, application-layer, widely applicable connection oriented mechanism for Internet client/server communications security. This paper gives a detailed technical analysis of the cryptographic strength of the SSL protocol. It is a cryptographic protocol which has been used broadly for making secure connection to a web server. SSL relies upon the use of dependent cryptographic functions to perform a secure connection. The first function is the authentication function which facilitates the client to identify the server and vice versa. There have been used, several other functions such as encryption and integrity for the imbuement of security. The most common cryptographic algorithm used for ensuring security is RSA. It still has got several security breaches that need to be dealt with. An improvement over this has been implemented in this paper. In this paper, a LBC has been proposed that switches from the domain of integers to the domain of bit stuffing to be applied to the first function of SSL that would give more secure communication. The introduction of bit stuffing will complicate the access to the message even after getting the access to the private key. So, it will enhance the security which is the inevitable requirement for the design of cryptographic protocols for secure communication.