This paper introduces Curve25519, a high-security elliptic-curve-Diffie-Hellman function designed for cryptographic applications. The function achieves record-setting speeds, computing 832,457 Pentium III cycles, and includes several side benefits such as free key compression, validation, and state-of-the-art timing-attack protection. The paper details the design and implementation of Curve25519, explaining its security, efficiency, and performance. It also discusses the choice of parameters, the arithmetic operations used, and the optimization techniques employed. The function is based on an elliptic curve over a prime field, with a prime number \( p = 2^{255} - 19 \) and a generator point \((9, \ldots)\). The paper covers the mathematical foundations, security analysis, and practical implementation, highlighting the function's advantages over previous implementations in terms of speed and security.This paper introduces Curve25519, a high-security elliptic-curve-Diffie-Hellman function designed for cryptographic applications. The function achieves record-setting speeds, computing 832,457 Pentium III cycles, and includes several side benefits such as free key compression, validation, and state-of-the-art timing-attack protection. The paper details the design and implementation of Curve25519, explaining its security, efficiency, and performance. It also discusses the choice of parameters, the arithmetic operations used, and the optimization techniques employed. The function is based on an elliptic curve over a prime field, with a prime number \( p = 2^{255} - 19 \) and a generator point \((9, \ldots)\). The paper covers the mathematical foundations, security analysis, and practical implementation, highlighting the function's advantages over previous implementations in terms of speed and security.