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The vast expansion of the Internet of Things (IoT) devices and related applications has bridged the gap between the physical and digital world. Unfortunately, security remains a major challenge and the lack of secure links have fueled the increased attacks on IoT devices and networks. Due to its inherent scalability, Public Key Infrastructure (PKI) is the well-known and classic approach to bring public-key certificate based security to IoT. Even though the standard X.509 explicit certificates can be viable solution, they are inefficient and too large for resource constrained IoT networks and therefore, smaller, faster and more efficient Elliptic Curve Qu Vanstone (ECQV) implicit certificates can be employed for establishing authenticated connections in IoT. Moreover, the existing certificate-based authentication proposals in standardized IoT networks have either been deployed at the transport or physical layers. Thus, these proposals fail to provide true end-to-end security to messages at the application layer in the presence of intermediate CoAP proxies. This challenging aspect is addressed in this proposal by focusing on the certificate-based authentication at the application layer to ensure true end-to-end security of messages. Additionally, IoT application layer security protocols like EDHOC lacks mechanism for authenticated distribution of public keys and thus, there is a need for lightweight authentication based cryptographic primitive for establishing secure key agreement in IoT. This paper introduces a design and implementation of a lightweight ECQV implicit certificate and use them for authenticated key exchange in EDHOC at the application layer. We also design a lightweight profile with a novel encoding mechanism for ECQV implicit certificate, called L-ECQV. To prove its viability, L-ECQV has been implemented and evaluated on Contiki operating system. Our evaluation results show that the proposed L-ECQV certificate approach reduces energy consumption by 27%, message overhead of EDHOC handshake by 52%, and shows improvements in certificate validation time. The security analysis demonstrates that proposed L-ECQV certificates for EDHOC protocol is secure against a number of attack vectors present in the IoT network. This novel combination of ECQV certificates with EDHOC key exchange leads to a secure and lightweight authenticated key agreement in IoT networks.