Application and Research of Algebraic Curve in Identity Authentication Security

In the field of information security, the security of identity authentication is crucial. We focus on the application of algebraic curves in the field of identity authentication security and explore their core content in depth. Starting from the basic theory of algebraic curves, we explain the group structure and discrete logarithmic properties of elliptic curves. Then, we conduct a detailed analysis of its application mode, including key generation, encryption and decryption, and digital signature processes based on elliptic curve cryptography (ECC). This is aimed at building a secure defense line for identity authentication. Meanwhile, by using algebraic curves to construct a zero - knowledge identity authentication protocol, we can achieve identity verification under privacy protection. Compared to traditional methods, this application has obvious advantages and can effectively resist quantum computing attacks, improving efficiency with shorter key lengths and faster computation speeds. However, in the process of application promotion, there are challenges such as the urgent need to complete mathematical theories, optimize algorithms, and ensure compatibility.Conduct research on the application of algebraic curves in identity authentication security, providing both theoretical foundations and practical guidance for promoting technological innovation in this field and further enhancing the security and reliability of identity authentication.