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Encyclopedia of Cryptography and Security pp 444–445Cite as

Factorization Circuits

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Related Concepts

Integer Factorization

Definition

Factorization circuits are circuits specially designed to factor integers.

Theory and Applications

There is a long history of special-purpose machines built for integer factorization, including the machine à congruences (Carissan, 1919), the bicycle chain sieve (Lehmer, 1926), the Georgia Cracker (Smith and Wagstaff, 1982), CAIRN 1 [10], CAIRN 2 [3] and CAIRN 3 [4]. There is also a long history of designs that have been published at various levels of detail but that have not been reported to have been built, including Quasimodo [8], TWINKLE [5], a mesh-sieving circuit [1], an ECM circuit [1], TWIRL [9], and SHARK [2].

Application-specific integrated electronic circuits are generally believed to be the most cost-effective current technology for large-scale well-funded cryptanalytic computations. Almost all recent proposals are factorization-specific very-large-scale integrated electronic circuits. TWINKLE is an exception, using optical...

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Recommended Reading

  1. Bernstein DJ (2001) Circuits for integer factorization: a proposal. URL: http://cr.yp.to/papers.html\#nfscircuit

  2. Franke J, Kleinjung T, Paar C, Pelzl J, Priplata C, Stahlke C (2005) Shark: a realizable special hardware sieving device for factoring 1024-bit integers. In: Rao JR, Sunar B (eds) Cryptographic hardware and embedded systems – CHES 2005, 7th international workshop, Edinburgh, 29 August–1 September 2005. Lecture notes in computer science, vol 3659. Springer, pp 119–130

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  3. Izu T, Kogure J, Shimoyama T (2007) CAIRN 2: an FPGA implementation of the sieving step in the number field sieve method. In: Paillier P, Verbauwhede I (eds) Cryptographic hardware and embedded systems – CHES 2007, 9th International workshop, Vienna, 10–13 September 2007. Lecture notes in computer science, vol 4727. Springer, pp 364–377

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  4. Izu T, Kogure J, Shimoyama T (2010) CAIRN: Dedicated integer factoring devices. In: International conference on network-based information systems, Takayama, pp 558–563

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  5. Lenstra AK, Shamir A (2000) Analysis and optimization of the TWINKLE factoring device. In: Preneel B (ed) Advances in cryptology – EUROCRYPT 2000, international conference on the theory and application of cryptographic techniques, Bruges, 14–18 May 2000. Lecture notes in computer science, vol 1807. Springer, pp 35–52

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  6. Lenstra AK, Shamir A, Tomlinson J, Tromer E (2002) Analysis of Bernstein’s factorization circuit. In: Zheng Y (ed) Advances in cryptology – ASIACRYPT 2002, 8th international conference on the theory and application of cryptology and information security, Queenstown, 1–5 December 2002. Lecture notes in computer science, vol 2501. Springer, pp 1–26

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  7. Lenstra AK, Tromer E, Shamir A, Kortsmit W, Dodson B, Hughes J, Leyland PC (2002) Factoring estimates for a 1024-bit RSA modulus. In: Laih C-S (ed) Advances in cryptology – ASIACRYPT 2003, 9th international conference on the theory and application of cryptology and information security, Taipei, 30 November–4 December 2003. Lecture notes in computer science, vol 2894. Springer, pp 55–74

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  8. Pomerance C, Smith JW, Tuler R (1988) A pipeline architecture for factoring large integers with the quadratic sieve algorithm. SIAM J Comput 17(2):387–403

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  9. Shamir A, Tromer E (2003) Factoring large numbers with the TWIRL device. In: Boneh D (ed) Advances in cryptology – CRYPTO 2003, 23rd Annual international cryptology conference, Santa Barbara, 17–21 August 2003, Proceedings. Lecture notes in computer science, vol 2729. Springer, pp 1–26

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  10. Shimoyama T, Izu T, Kogure J (2005) Implementing a sieving algorithm on a dynamic reconfigurable processor (extended abstract). In: Special-purpose hardware for attacking cryptographic systems – SHARCS’05, Paris. http://www.hyperelliptic.org/tanja/SHARCS/talks/SHARCS2005paper-shimoyama.pdf

  11. Wiener MJ (2004) The full cost of cryptanalytic attacks. J Cryptol 17:105–124

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Chicago, Chicago, Illinois, USA

    Daniel J. Bernstein

Authors
  1. Daniel J. Bernstein
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

  2. Center for Secure Information Systems, George Mason University, Fairfax, VA, 22030-4422, USA

    Sushil Jajodia

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Bernstein, D.J. (2011). Factorization Circuits. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_447

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