Elliptic Curve Cryptography (ECC) is a contemporary form of asymmetric key cryptography widely employed in critical security domains, including encryption, decryption, and authentication. Its utilization has become increasingly prevalent in the Vital Information Infrastructure (IIV) to ensure secure communication and data exchange over time, driven by technological advancements and the growing size of data. This paper aims to provide a comparative relative efficiency analysis of ECC usage in the IIV, focusing on key aspects such as time efficiency, the relationship between curve size and security strength level, and the resilience of different curve sizes against Brute-Force attacks, Pollard Rho attacks, and Elliptic Curve Discrete Logarithm Problem (ECDLP) attacks. The results demonstrate that SECP521R1 features the largest key size and slowest key exchange time. Notably, SECP521R1 displays the highest level of resistance against various attacks, making it the most robust curve in terms of security.