In modern cryptography, improving the cryptographic security of Zero-Knowledge Proofs (ZKP) has become a compelling trend. Traditional models like the zk-SNARK and zk-STARK has shown strong security but are accompanied by the inherent issues of computational complexity and proof size. This work presents the Algebraic Zero-Knowledge Proof (AZKP) framework, using algebraic structures and integration of elliptic curves to optimize proof creation and verification. The suggested approach fills in key gaps found in the current methodologies, such as huge computational overhead and enormous proof sizes. Prime factorization in algebraic groups and ring homomorphisms of the AZKP framework is used to achieve small proof size without sacrificing computational efficiency. Comparing AZKP with zk-SNARK and zk-STARK models, experimental evaluation was applied to four critical performance metrics. generation time of proof, verification time, size of proof, and computational overhead. Results show that the AZKP is able to make a 48% decrease in proof generation duration and 20% increase in verification speed in comparison to zk-SNARK. Also, AZKP incurred lower computational cost than zk-STARK, with a proof size that is manageable. These results highlight the prospect of AZKP in cryptographic use where high-speed low-latency verification operations are desired. Further research will integrate AZKP in blockchain environments in order to increase real-time transaction validation.