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Secure Socket Layer (SSL) is a cryptographic protocol widely used to make a secure connection to a Web server. SSL uses three interdependent cryptographic functions to perform a secure connection. The first function is authentication. It is used to allow the client to identify the server and optionally allow the server to identify the client. The most common cryptographic algorithm used for this function is RSA. If we double the key length in RSA to have more secure communication, then it is known that the time needed for the encryption and decryption will be increased approximately eight times. In this paper, we propose a modification of RSA from the domain of integers to the domain of Gaussian arithmetic to be applied to the first function of SSL that would give more secure communication. This modification would use only double the time needed for the usual implementation of RSA with key size of 1024 bits.