Smartphone usage is increasing rapidly generating an era of nomadic computer users, providing easy access to information anywhere, anytime. As a result, smartphone users do tasks like, accessing email, browsing web, e-banking etc on their mobile devices. Hence, many mobile apps today need secure communications between the smartphone and the end server. The most common method for protecting web traffic is the Transport Layer Security (TLS) protocol. However, cryptographic operations such as public key crytpo stress the CPU. In this paper, we evaluate whether offloading the asymmetric key exchange of the TLS protocol to a cloud instance could bring some benefit from the mobile device point of view. In particular, we designed a possible solution (named CaT) for offloading the TLS asymmetric key exchange, and conducted an extensive experimental analysis. The results show that the CaT is actually less energy-efficient than performing the normal full TLS handshake on a smartphone. We discuss the motivations of this results, as well as possible directions to improve the obtained results.