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Description: ABSTRACT Quantum algorithms for optimisation and cryptography offer exponential speedups over classical algorithms and can solve difficult computing problems. This literature review covers important quantum algorithms for optimisation and cryptography investigations from 2008 to 2017. various studies will help us understand quantum algorithm development, methodologies, and applications in various disciplines. Grover's method, a fast quantum search algorithm, and Shor's algorithm, known for its discrete logarithm responses and polynomial-time factorization, are examined first. We address these algorithms' consequences for database search, combinatorial optimisation, cryptanalysis, and cryptography. The study also discusses recent advances like the quantum approximate optimisation algorithm (QAOA), which uses classical and quantum approaches to handle combinatorial optimisation problems. Quantum random walks are employed in graph problems and search algorithms. We examine the quantum algorithm for linear systems of equations in data fitting, optimisation, and simulation. This review focuses on cryptographic algorithms, particularly hidden shift and hidden subgroup concerns. Quantum cryptanalysis uses quantum phenomena to break cryptography systems. We also study quantum counting algorithms, which count database patterns or solutions efficiently. The literature review evaluates each algorithm's benefits, cons, and potential uses. We discuss how quantum algorithms speed up processing, influence traditional approaches, and are difficult to implement and run. Noise reduction and error correction improve quantum computing accuracy and reliability. It is a useful tool for academics and industry professionals who want to use quantum computing to strengthen cryptography protocols and solve challenging optimisation challenges. The knowledge gathered from this review helps with the continuous research and development of quantum algorithms and their potential application in many different domains of business and science. Keywords: Auctions, Market Design, Resource Allocation, Multi-Agent Systems, Algorithmic Game Theory.