Shor's Algorithm

Theoretical analysis of Shor's quantum factorization algorithm and its implications for modern cryptography and computational complexity

In-depth analysis of Shor's algorithm for integer factorization, focusing on the quantum order-finding subroutine and quantum phase estimation techniques. The study examines the classical reduction from factorization to the order-finding problem and details the quantum circuit implementation using controlled-unitary operations and the Quantum Fourier Transform. Performance analysis demonstrates the algorithm's polynomial-time complexity compared to the sub-exponential complexity of classical algorithms. The project explores real-world implementation challenges, including noise and error accumulation in current quantum hardware, and evaluates the cryptographic impact on RSA and ECC security. Includes computational benchmarks and complexity comparisons.