This paper researches on cloud computing based-on authentication for the security verification using Elliptic Curve Digital Signature Algorithm (ECDSA). The study structure simply focuses on the communication interaction of the Cloud providing communities and a pictured Smartphone user. The Elliptic Curve Cryptosystem is envisioned as the basic framework for data transmission security in the paper that proffers solution. We reviewed various literatures on the elliptic curves and applied the cyclic elliptic curve group based-on finite field modulo q. With the theory established, we applied the Elliptic Curve Digital Signature Algorithm as developed by ElGamal to carry out some exemplified computational sequences. The detail of the computational algorithm was done in Mathematica as given by (Jouko, 2011) with no modifications except in its translational interpretation to fit into cloud computing security environment as maybe applied by the cloud providing communities to accelerate and meeting the needs of the users on-demand as a service with embedding security consideration. Our contribution to the cloud computing environment is brought about through computational experimental activities as exemplified in this field for interactive security awareness through digital signature authenticating mechanism.

This paper dwells theoretically on the application of Elliptic Curves Cryptography (ECC) for the web and Cloud and Cloud Computing environments. ECC is still in its embryonic utility formation unlike the RSA that has gained much ground in its application on the web security as it offers extensive applications to the various aspects of authentications, ecommerce and mobile phones; but it consumes a lot of power that may not be useful to Smartphones in the context of Mobile cloud computing. Notwithstanding, there is some remarkable deployment of ECC for web security currently ongoing This paper establishes the position and role of ECC in the web computing environment. A general introduction to the web and security threats emerging from the use of the web is discussed. Different types of security protocols on the web (IPSec, SSL/TLS and SSH) are discussed. The paper expounds how ECC is deployed on the SSL/TLS protocol-the dominant security protocol on the web-to contain the adversaries. Moreover, we made an effort to discover why people may make the choice of ECC over other cryptographic systems like RSA and discrete logarithm cryptographic systems. Finally, with the hype in mobile Cloud Computing, the paper discusses the security implementations how ECC is applied on Digital Signature