Enhancing Post-Quantum Cryptography: Exploring Mathematical Foundations and Comparative Analysis of Different Cryptographic Algorithm

Abstract

Abstract: This research paper surveys the landscape of cryptography, encompassing historical origins and modern applications. Beginning with foundational concepts, it explores encryption, decryption, ciphers, and keys, spanning symmetric and asymmetric cryptography. Historical context unfolds, tracing cryptography from ancient Egyptian hieroglyphs to Julius Caesar's cipher. The study then transitions to contemporary subjects. Elliptic Curves and Cryptography are investigated, showcasing their significance in secure communication, demonstrating ECC key exchange and AES-GCM encryption using python and Comparative analysis of ECC, RSA, and Diffie-Hellman sheds light on their performance. Zero-Knowledge Proofs are introduced as tools for privacy-preserving verification followed by an exploration of various Zero-Knowledge Proof (ZKP) protocols. By presenting practical implementation examples using Python, the paper illustrates how these proofs can be applied in real-world scenarios. Random Number Generation is examined and distinction between pseudorandom number generators (PRNGs) and cryptographically secure PRNGs (CSPRNGs) is emphasized conducting a thorough comparative analysis of PRNGs and CSPRNGs, considering factors like correlation, independence, periodicity, and entropy. Furthermore, the section evaluates the performance of different random number generation techniques. Fully Homomorphic Encryption emerges as a groundbreaking concept, discussing its mathematical properties, practical implementation, parameter selection, and optimization techniques enabling computation on encrypted data. Cryptographic Secret Sharing Schemes are explored for secure information distribution. The paper concludes by delving into the Chinese Remainder Theorem's applications within modern cryptographic protocols, particularly in RSA decryption and the integration factorization process of the RSA public key cryptosystem. It also provides a comprehensive overview of the theoretical foundations of primality testing, a pivotal aspect of the RSA algorithm. Overall, this research paper provides a comprehensive exploration of cryptography's historical context, core concepts, advanced techniques, and practical implementations, offering valuable insights into the realm of secure communication