Primality testing is a fundamental problem in computational number theory and modern cryptography, where the efficient generation and verification of large prime numbers are essential for the security of public-key cryptosystems. Among the numerous primality testing techniques, elliptic curve-based methods have emerged as one of the most powerful approaches due to their strong mathematical foundation and practical efficiency. This paper presents the mathematical foundations of elliptic curve primality tests by reviewing the algebraic structure of elliptic curves over finite fields, the group law, Hasse's theorem, and the properties of the Frobenius endomorphism that underpin these algorithms. The study examines the principles of major elliptic curve primality proving methods, including the Goldwasser–Kilian algorithm and the Atkin–Morain Elliptic Curve Primality Proving (ECPP) algorithm. Furthermore, elliptic curve-based techniques are compared with classical primality tests such as the Fermat, Solovay–Strassen, Miller–Rabin, and AKS algorithms in terms of computational complexity, accuracy, and practical applicability. The analysis demonstrates that elliptic curve methods provide a mathematically rigorous and computationally efficient framework for proving the primality of large integers, making them particularly suitable for cryptographic applications requiring certified prime numbers. The presented mathematical framework serves as a comprehensive reference for understanding the theoretical principles and practical significance of elliptic curve primality tests.