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Abstract Security in the world of the internet has become very important in all aspects of social life. One of the methods of securing the information on the internet is public-key cryptography or asymmetric cryptography. Public key cryptography is not only a process of encrypting information, it also provides confidentiality, data integrity and authentication. RSA and ElGamal are two very important public-key cryptosystems. These two cryptosystems are also used for digital signature scheme because of their high level of security. In this paper, we discussed RSA algorithm in details with example. RSA is one of the most widely used public-key cryptography in various application. Then we discussed the ElGamal algorithm in details with example. ElGamal public-key cryptography is also used in many applications nowadays. In the next section, we discussed digital signature using RSA algorithm and ElGamal algorithm as digital signature is one of the important application of public-key cryptography. Therefore, Reader will have a good understanding of public-key cryptography. In this paper we have also included a relative study between RSA and ElGamal.