In the Internet of Things (IoT) environment, more types of devices than ever before are connected to the internet to provide IoT services. Smart devices are becoming more intelligent and improving performance, but there are devices with little computing power and low storage capacity. Devices with limited resources will have difficulty applying existing public key cryptography systems to provide security. Therefore, communication protocols for various kinds of participating devices should be applicable in the IoT environment, and these protocols should be lightened for resources-restricted devices. Security is an essential element in the IoT environment, so for secure communication, it is necessary to perform authentication between the communication objects and to generate the session key. In this paper, we propose two kinds of lightweight authentication and key agreement schemes to enable fast and secure authentication among the objects participating in the IoT environment. The first scheme is an authentication and key agreement scheme with limited resource devices that can use the elliptic curve Qu-Vanstone (ECQV) implicit certificate to quickly agree on the session key. The second scheme is also an authentication and key agreement scheme that can be used more securely, but slower than first scheme using certificateless public key cryptography (CL-PKC). In addition, we compare and analyze existing schemes and propose new schemes to improve security requirements that were not satisfactory.