Quantum Computing and Cryptography

Quantum computing endangers classical cryptography via two main avenues: Shor’s al-gorithm reduces integer factorization to polynomial time, and Grover’s algorithm yields a quadratic speed-up for unstructured key search. We develop rigorous mathematical treatment of both algorithms, enrich the literature survey with detailed complexity and resource analyses, and extend the development section to include quantum er-ror-correction overhead, resource estimates for breaking 2048-bit RSA, and advanced as-ymptotic bounds. In Discussion, we propose lattice-, code-, multivariate-, and hash-based post-quantum schemes, analyze their security margins against quantum attacks, and out-line standardized migration strategies. We include three illustrative figures and 35+ se-quential IEEE citations.