Elliptic Curve Cryptography (ECC) achieves the same 128-bit security level as 3072-bit RSA but with significantly smaller key sizes. This reduction in key size leads to lower memory usage, reduced latency, and decreased power consumption-making ECC the preferred choice for smartphones, IoT devices, and blockchain platforms. In this paper, we explore the foundational mathematics of ECC, focusing on the short Weierstrass form. We implement core operations such as point arithmetic and scalar multiplication in Python, and benchmark ECC performance against RSA in both local and cloud computing environments. Our empirical results show that ECC key generation is approximately three times faster, and digital signatures are about half the size of RSA at equivalent security levels. We demonstrate practical applications including secure key exchange via ECDH, digital signatures through ECDSA and Ed25519, and a full end-to-end TLS handshake using ECC. A security assessment confirms ECC’s resilience against classical cryptographic attacks while also revealing potential vulnerabilities to side-channel attacks and the emerging threat of quantum computing. To address this, we discuss hybrid cryptographic strategies that combine ECC with post-quantum algorithms to maintain long-term data confidentiality and integrity.