An efficient practical alternative to ECC by Pell curve cryptography with a new vision

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The main intention of the research is to reduce the key size for ECC using Pell curve group of finite order 2p while maintaining the same hardness as ECC. In order to achieve this we establish a robust path between the two distinct fields of Pell curve and cryptography to withstand the security challenges in the modern digital environment. There are two main objectives. The first objective is finding all the solutions to the Pell equation in the 2D plane and constructing these solutions as an algebraic structure, which is called the Pell Curve Group (PCG). Then we identify a specific Pell equation that will give a Pell curve group of exact finite order and will be the underlying work infrastructure to implement the security protocols effectively. The second and primal objective is to design a novel Diffie-Hellman Model blind key agreement protocol and cryptosystem on this platform. The significance of PCG is that the key sizes can be reduced to half of the key sizes of Elliptic Curve Cryptography (ECC) with a high level of security. We also investigated that by implanting the Brahmagupta identity as a binary operation, the computations became faster than in ECC. We strongly believe that this algorithm would be the fastest among all existing so far, and we concluded analytically using Pollard’s Rho method. The discrete logarithm problem over a Pell curve group underpins and attests to the security of the proposed cryptographic protocols. In support of this notion, comparative analysis and efficiency are also proficiently presented in terms of security over Pell curve cryptography. The proposed algorithm, which is secure and quicker in execution, can effectively handle time decisive online transactions like airline, train reservations, banking, and other situations.