Quantum Shor's Algorithm vs. Classical General Number Field Sieve (GNFS) Approach for Large Integer Factorization

Quantum integer factorization is a groundbreaking field that explores the use of quantum algorithms, notably Shor's algorithm, to efficiently factorize large integers into their prime components. Its potential implications on modern encryption techniques, such as the Rivest-Shamir-Adleman (RSA) algorithm, are of considerable interest. This paper's central aim is to contrast the computational time and implementation of the Integer Factorization Problem (IFP) between classical and quantum computers. The inner workings of Quantum Shor's algorithm are broken down step by step, and the estimated processing time is contrasted with the classical General Number Field Sieve (GNFS) approach. Given the exponential speedup provided by quantum computing, this comparison helps assess Shor's algorithm's possible advantage over GNFS in factoring large numbers. Direct comparisons with GNFS are currently difficult due to Shor's algorithm's reliance on large-scale quantum computer development for practical execution speeds. This paper focuses on solving integer factorization up to $2^{n}$ bits.