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Strategies in Filtering in the Number Field Sieve

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Algorithmic Number Theory (ANTS 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1838))

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Abstract

A critical step when factoring large integers by the Number Field Sieve [8] consists of finding dependencies in a huge sparse matrix over the field \(\mathbb{F}_{2}\), using a Block Lanczos algorithm. Both size and weight (the number of non-zero elements) of the matrix critically affect the running time of Block Lanczos. In order to keep size and weight small the relations coming out of the siever do not flow directly into the matrix, but are filtered first in order to reduce the matrix size. This paper discusses several possible filter strategies and their use in the recent record factorizations of RSA-140, R211 and RSA-155.

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References

  1. Boender, H.: Factoring Large Integers with the Quadratic Sieve. PhD thesis, Rijksuniversiteit Leiden (1997)

    Google Scholar 

  2. Buhler, J.P., Lenstra Jr., H.W., Pomerance, C.: Factoring integers with the number field sieve. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554, pp. 50–94. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  3. Cavallar, S., Dodson, B., Lenstra, A.K., Leyland, P., Lioen, W., Montgomery, P.L., Murphy, B., te Riele, H., Zimmermann, P.: Factorization of RSA-140 using the number field sieve. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 195–207. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  4. Cavallar, S., Dodson, B., Lenstra, A.K., Leyland, P., Lioen, W., Montgomery, P.L., te Riele, H., Zimmermann, P.: 211-digit SNFS factorization (April 1999), Available from ftp://ftp.cwi.nl/pub/herman/NFSrecords/SNFS-211

  5. Cavallar, S., Dodson, B., Lenstra, A.K., Lioen, W., Montgomery, P.L., Murphy, B.H., te Riele, K., Aardal, J., Gilchrist, G., Guillerm, P., Leyland, J., Marchand, F., Morain, A., Muffett, C., Putnam, C., Putnam, P.: Zimmermann. Factorization of a 512-bit RSA modulus. Submitted to Eurocrypt (2000)

    Google Scholar 

  6. Cowie, J., Dodson, B., Elkenbracht-Huizing, R.-M., Lenstra, A.K., Montgomery, P.L., Zaye, J.: A world wide number field sieve factoring record: on to 512 bits. In: Kim, K.-c., Matsumoto, T. (eds.) ASIACRYPT 1996. LNCS, vol. 1163, pp. 382–394. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  7. Denny, T.F.: Solving large sparse systems of linear equations over finite prime fields. Transparencies of a lecture of the Cryptography Group at CWI (May 1995)

    Google Scholar 

  8. Elkenbracht-Huizing, R.-M.: An implementation of the number field sieve. Experimental Mathematics 5(3), 231–253 (1996)

    MATH  MathSciNet  Google Scholar 

  9. Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Annals of the History of Computing 7(1), 43–57 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Neukirch, J.: Algebraische Zahlentheorie. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  11. Knuth, D.E.: The Stanford GraphBase: A Platform for Combinatorial com- puting. Addison-Wesley, Reading (1993)

    Google Scholar 

  12. Knuth, D.E.: Sorting and Searching, The Art of Computer Pro- gramming, 2nd edn., vol. 3. Addison-Wesley, Reading (1998)

    Google Scholar 

  13. LaMacchia, B.A., Odlyzko, A.M.: Solving large sparse linear systems over finite fields. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 109–133. Springer, Heidelberg (1991)

    Google Scholar 

  14. Lang, S.: Algebraic Number Theory. Springer, Heidelberg (1986)

    MATH  Google Scholar 

  15. Montgomery, P.L.: Square roots of products of algebraic numbers. In: Gautschi, W. (ed.) Mathematics of Computation 1943–1993: a Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics, vol. 48, pp. 567–571. American Mathematical Society (1994)

    Google Scholar 

  16. Montgomery, P.L.: A block Lanczos algorithm for finding dependencies over GF(2). In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 106–120. Springer, Heidelberg (1995)

    Google Scholar 

  17. Nguyen, P.: A Montgomery-like square root for the number field sieve. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 151–168. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  18. Pollard, J.M.: The lattice sieve. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554, pp. 43–49. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  19. Pomerance, C., Smith, J.W.: Reduction of huge, sparse matrices over finite fields via created catastrophes. Experimental Mathematics 1(2), 89–94 (1992)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centrum voor Wiskunde en Informatica, Postbus 94079, 1090 GB, Amsterdam, The Netherlands

    Stefania Cavallar

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  1. Stefania Cavallar
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Editors and Affiliations

  1. Mathematisch Instituut, Universiteit Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    Wieb Bosma

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© 2000 Springer-Verlag Berlin Heidelberg

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Cavallar, S. (2000). Strategies in Filtering in the Number Field Sieve. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_11

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  • DOI: https://doi.org/10.1007/10722028_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67695-9

  • Online ISBN: 978-3-540-44994-2

  • eBook Packages: Springer Book Archive

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