Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm which runs on a quantum computer) for integer factorization discovered in 1994. Informally it solves the following problem: Given an integer N, find its prime factors.
On a quantum computer, to factor an integer N, Shor's algorithm runs in polynomial time (the time taken is polynomial in log N, which is the size of the input). Specifically it takes time O((log N)3), demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is thus in the complexity class BQP. This is exponentially faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time -- about O(e(log N)1/3 (log log N)2/3). The efficiency lies in the efficiency of the quantum Fourier transform, and modular exponentiation by squarings.
Given a quantum computer with a sufficient number of qubits, Shor's algorithm can be used to break the widely used public-key cryptography scheme known as RSA. RSA is based on the assumption that factoring large numbers is computationally infeasible. So far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor in polynomial time. However, Shor's algorithm shows that factoring is efficient on a quantum computer, so an appropriately large quantum computer can break RSA. It was also a powerful motivator for the design and construction of quantum computers and for the study of new quantum computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.
In 2001, Shor's algorithm was demonstrated by a group at IBM, who factored 15 into 3 × 5, using an NMR implementation of a quantum computer with 7 qubits. However, some doubts have been raised as to whether IBM's experiment was a true demonstration of quantum computation, since no entanglement was observed. Since IBM's implementation, several other groups have implemented Shor's algorithm using photonic qubits, emphasizing that entanglement was observed.
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