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Concept# Shor's algorithm

Summary

Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. It is one of the few known quantum algorithms with compelling potential applications and strong evidence of superpolynomial speedup compared to best known classical (that is, non-quantum) algorithms. On the other hand, factoring numbers of practical significance requires far more qubits than available in the near future. Another concern is that noise in quantum circuits may undermine results, requiring additional qubits for quantum error correction.
Shor proposed multiple similar algorithms solving the factoring problem, the discrete logarithm problem, and the period finding problem. "Shor's algorithm" usually refers to his algorithm solving factoring, but may also refer to each of the three. The discrete logarithm algorithm and the factoring algorithm are instances of the period finding algorithm, and all three are instances of the hidden subgroup problem.
Shor's algorithm allows to factor an integer on a quantum computer in polylogarithmic time, meaning that the running time of the algorithm is polynomial in . Specifically, it takes quantum gates of order using fast multiplication, or even utilizing the asymptotically fastest multiplication algorithm currently known due to Harvey and Van Der Hoven, thus demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class BQP. This is significantly faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time: .
If a quantum computer with a sufficient number of qubits could operate without succumbing to quantum noise and other quantum-decoherence phenomena, then Shor's algorithm could be used to break public-key cryptography schemes, such as
The RSA scheme
The Finite Field Diffie-Hellman key exchange
The Elliptic Curve Diffie-Hellman key exchange
RSA is based on the assumption that factoring large integers is computationally intractable.

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