CHAOS TO CLARITY:CRYPTOGRAPHIC HASHING AND APPROXIMATE SIMILARITY SEARCH VIA COLLATZ CONJECTURE AND KERNEL METHODS

Abstract

In an era where data security and rapid information retrieval underpin nearly every digital interaction, this thesis embarks on a journey that bridges the elegance of mathematics with the urgency of real-world challenges. At its heart lies a seemingly simple mathematical puzzle, the Collatz Conjecture, whose unpredictable nature, when fused with the deterministic chaos of non-linear systems, unlocks powerful cryptographic possibilities. This work introduces a robust cryptographic framework that harnesses the modified Collatz sequences and chaotic dynamics to design a novel hash function. The function exhibits exceptional randomness, uniform distribution, and sensitivity to initial conditions, essential traits for securing modern digital systems. Through rigorous statistical validation and comparative analysis, the proposed algorithm consistently outperforms conventional hashing standards such as SHA-2 and SHA-3 in terms of collision resistance, avalanche effect, and computational speed. Building on this foundation, the research advances into the domain of image encryption, where a pixel-level encryption technique is crafted. Leveraging the same chaotic-Collatz synergy, the encryption scheme ensures pixel decorrelation, high entropy, and strong resistance against statistical, differential, and brute-force attacks. The final chapter of this journey tackles the high-dimensional complexity of big data. By embedding the hash function into a Gaussian-kernel-enhanced locality-sensitive hashing (LSH) framework, the thesis delivers an efficient and accurate similarity search solution. This hybrid LSH model preserves semantic similarity while drastically reducing retrieval time. Tested on standard text and Image datasets like SIFT1M, GIST1M, and MS-MARCO, the proposed method outpaces state-of-the-art algorithms such as FAISS, offering higher recall at reduced computational cost. Together, these contributions form a cohesive narrative: a mathematical conjecture, long seen as theoretical curiosity, is reimagined as a cornerstone of practical, scalable, and future-proof cryptographic design. This thesis not only elevates the role of mathematical structures in data security but also lays the groundwork for secure image communication, cloud storage protection, and real-time data retrieval systems in the age of AI and big data

Date of Award	2025
Original language	American English
Awarding Institution	HBKU College of Science and Engineering