Implementation of Shor's Algorithm and Its Demonstrated Quantum Efficiency

Abstract

The unique principles of quantum physics introduce an element of uncertainty into our previously deterministic world. Quantum computing, derived from quantum physics, enables the representation of a qubit in both |0> and |1> states, thus encoding more information and unleashing a realm of new possibilities. This paper conducts an analysis of the classical RSA encryption protocol and its vulnerability to the quantum algorithm—Shor's algorithm. Furthermore, this paper rigorously establishes the mathematical theories required for the analysis. Finally, the algorithm is thoroughly explored, and the potential efficiencies of quantum computing are discussed. As the study of quantum computing deepens, its application is poised to surpass that of traditional computers in various fields, leaving a profound impact.