Cryptographic Applications of Pell-Like Equation

This project investigates Pell’s Equation and its generalization, the Pell-like Equation, highlighting their roles in cryptography. These equations form the basis for defining L-Groups, which are specialized cryptographic groups used to develop secure protocols such as the Diffie-Hellman key exchange and the ElGamal cryptosystem. By leveraging the mathematical properties of L-Groups, these protocols enable two parties to create a shared secret key or encrypt messages without directly sharing sensitive data, relying instead on the difficulty of the discrete logarithm problem for security. Through structured analysis, the project demonstrates how these cryptographic applications protect data confidentiality and integrity by translating complex mathematical problems into practical security tools. The work illustrates the deep connections between number theory and cryptography, showcasing Pell’s Equation as a valuable framework for building secure communication protocols and encouraging further research in mathematically grounded cryptographic methods.