Binary extension field-based Elliptic Curve Cryptography (ECC) offers secure and efficient capabilities to resource-constrained systems such as embedded systems and IoT. In this paper, a specific research is provided on ECC as characterized in finite fields F2m, and the mathematical structure, arithmetic and cryptographic resiliency are discussed. The representations of polynomials and normal basis are also discussed and some of the basic field operations like addition, multiplication, and inversion are carried out. It also examines the elliptic curve constructions in binary fields, in which the different kinds of curves have been supersingular and non-supersingular. Good group law arithmetic to compute point addition and point doubling are considered to optimize the performance of scalar multiplication. Lower computational complexity and improved energy efficiency, over prime field implementations, has been demonstrated experimentally. The results prove the suitability of binary field ECC in high-speed low-power cryptography applications and simultaneously, they are very resistant to classical cryptanalysis attacks and have high security assurance.