A deep dive into the mathematical security foundations of Elliptic Curve Cryptography (ECC), focusing on the Elliptic Curve Discrete Logarithm Problem (ECDLP). Explains why ECDLP is exponentially hard compared to the ordinary discrete log problem, why ECC keys need roughly twice as many bits as symmetric keys, and how the embedding degree affects security against extension-field attacks. Includes a practical table mapping security levels to subgroup sizes, extension field sizes, and embedding degrees, plus guidance on choosing prime embedding degrees and the computational cost of finding pairing-friendly curves.
