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Internet has revolutionized the data communication systems. It provides platform to get the information exchanged quickly amongst the communicating parties at the same time it also provides opportunity to adversary to attack on unsecured information. In order to provide confidentiality, integrity and authentication services to unsecured information while transit or static, cryptographic techniques are used. This paper analyses the security strength of two popular and practical public-key cryptography techniques RSA (Rivest Shamir Adleman) and ECC (Elliptic Curve Cryptography). RSA is considered first generation public-key cryptography, which is very popular since its inception while ECC is gaining popularity recently. The security of the RSA cryptosystem is based on the Integer Factorization Problem (IFP) and the security of ECC is based on elliptic curve discrete logarithm problem (ECDLP). The main attraction of ECC over RSA is that the best known algorithm for solving the ECDLP takes full exponential time while to solve IFP of RSA takes sub-exponential time. This means that significantly smaller parameters can be used in ECC than RSA, with equivalent levels of security. For example to achieve 112 bits of security level, RSA algorithm needs key size of 2048 bits, while ECC needs key size of 224-255 bits.