Thanks to recent advancements in biomedical sensors, wireless networking technologies, and information networks, traditional healthcare methods are evolving into a new healthcare infrastructure known as the Internet of Medical Things (IoMT). It enables patients in remote areas to obtain preventative or proactive healthcare services at a cheaper cost through the ease of time-independent interaction. Despite the many benefits of IoMT, the ubiquitously linked devices offer significant security and privacy concerns for patient data. In the literature, several multi-message and multi-receiver signcryption schemes have been proposed that use traditional public-key cryptography, identity-based cryptography, or certificateless cryptography methods to securely transfer patient health-related data from a variety of biomedical sensors to healthcare professionals. However, certificate management, key escrow, and key distribution are all complications with these methods. Furthermore, in terms of IoMT performance and privacy requirements, they are impractical. This article aims to include edge computing into an IoMT with secure deployment employing a multi-message and multi-receiver signcryption scheme to address these issues. In the proposed method, certificate-based signcryption and hyperelliptic curve cryptography (HECC) have been coupled for excellent performance and security. The cost study confirms that the proposed scheme is better than the existing schemes in terms of computational and communication costs.