Among the increasing evolution of IoT devices, practical applications need reliable secure protocols to communicate with each other. A major issue for modern cryptosystems is an implementation of secure and trustworthy mechanisms to rely on. A side-channel attack against these cryptosystems may overturn the guarantee of security against conventional cyber-attacks. Elliptic curve cryptography is public-key cryptography based on elliptic curves, and one of the well-known curves is Curve25519 which is used for TLS protocols as a recommended curve. This curve is mainly implemented on limited resource devices such as microcontrollers. However, this curve poses a weakness for low-order points during a Diffie-Hellman key exchange is employed. This research demonstrates possible exploitation of a threat of order 8 rational points of Curve25519 and shows results of the side-channel attacks using order 8 rational points on an embedded system. The results indicate the order 8 rational points might be applied to key extraction as attacker sides.