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Encyclopedia of Algorithms pp 1651–1652Cite as

Quantum Algorithm for Factoring

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Years and Authors of Summarized Original Work

  • 1994; Shor

Problem Definition

Every positive integer n has a unique decomposition as a product of primes \(n = p_{1}^{e}{}_{1}\cdots p_{k}^{e}{}_{k}\), for prime number p i , and positive integer exponent e i . Computing the decomposition \(p_{1},e_{1},\ldots ,p_{k},e_{k}\) from n is the factoring problem.

Factoring has been studied for many hundreds of years, and exponential time algorithms for it were found to include trial division, Lehman’s method, Pollard’s ρ method, and Shank’s class group method [1]. With the invention of the RSA public-key cryptosystem in the late 1970s, the problem became practically important and started receiving much more attention. The security of RSA is closely related to the complexity of factoring, and in particular, it is only secured if factoring does not have an efficient algorithm. The first subexponential-time algorithm is due to Morrison and Brillhard [4] using a continued fraction algorithm. This was...

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Recommended Reading

  1. Cohen H (1993) A course in computational algebraic number theory. Graduate texts in mathematics, vol 138. Springer, Berlin/Heidelberg/New York

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  2. Lenstra A, Lenstra H (eds) (1993) The development of the number field sieve. Lecture notes in mathematics, vol 1544. Springer, Berlin

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  3. Lenstra AK, Lenstra HW Jr, Manasse MS, Pollard JM (1990) The number field sieve. In: Proceedings of the twenty second annual ACM symposium on theory of computing, Baltimore, 14–16 May 1990, pp 564–572

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  4. Morrison M, Brillhart J A method of factoring and the factorization of F7

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  5. Pomerance C Factoring. In: Pomerance C (ed) Cryptology and computational number theory. Proceedings of symposia in applied mathematics, vol 42. American Mathematical Society, Providence, p 27

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  6. Shor PW (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput 26:1484–1509

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, The Pennsylvania State University, University Park, State College, PA, USA

    Sean Hallgren

Authors
  1. Sean Hallgren
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Corresponding author

Correspondence to Sean Hallgren .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Hallgren, S. (2016). Quantum Algorithm for Factoring. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_307

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