Shor's algorithm solves factoring problems exponentially faster than classical algorithms by leveraging the principles of quantum computing. Specifically, it takes advantage of quantum superposition and entanglement to perform calculations that would otherwise be infeasible for classical computers. While classical algorithms such as the general number field sieve can take an exponential amount of time to factor large integers, Shor's algorithm reduces this complexity to polynomial time, specifically (O((\log N)^2 (\log \log N) (\log \log \log N))). This means that as the size of the integer increases, the time it takes to find its factors grows much more slowly with Shor’s algorithm compared to classical methods.
The core of Shor's algorithm involves two main tasks: finding the period of a specific function and using this period to deduce the factors of the number. The algorithm first reduces the problem of factoring into a simpler problem of finding the period of a function related to modular exponentiation. Using quantum bits, or qubits, Shor's algorithm can evaluate this function simultaneously for many inputs, which classical bits cannot do. This parallelism allows it to identify the period of the function much faster than any classical approach. Once the period is determined, the algorithm uses mathematical properties to find the factors, which can often lead to the resolution of the factoring problem efficiently.
An example to illustrate this is factoring the number 15. For a classical method, you might try all combinations of factors, spent considerable time doing so. However, using Shor's algorithm, you can exploit quantum computation to find the period of the function much faster, leading to identifying that 15 can be factored into 3 and 5 with far fewer computations. As a result, Shor's algorithm showcases the potential of quantum computing to outperform traditional methods, especially for tasks like factoring large integers, which has significant implications for cryptography and computer security.