Article

Experiments on the linear algebra step in the number field sieve

Authors:

Kazumaro Aoki,

Takeshi Shimoyama,

Hiroki UedaAuthors Info & Claims

IWSEC'07: Proceedings of the Security 2nd international conference on Advances in information and computer security

Pages 58 - 73

Published: 29 October 2007 Publication History

Abstract

This paper shows experimental results of the linear algebra step in the number field sieve on parallel environment with implementation techniques. We developed an efficient algorithm that shares the sum of vectors in each node, and the network structure among the nodes only requires to include a ring. We also investigated the construction of a network for the linear algebra step. The construction can be realized through switches and network interface cards, whose prices are not expensive. Moreover, we investigated the implementation of the linear algebra step using various parameters. The implementation described in this paper was used for the integer factoring of a 176 digit number by GNFS and a 274 digit number by SNFS.

References

[1]

Lenstra, A.K., Lenstra Jr., H.W. (eds.): The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993).

Google Scholar

[2]

Kleinjung, T.: rsa200 (2005), http://www.crypto-world.com/announcements/rsa200.txt

Google Scholar

[3]

Silverman, R.D.: A cost-based security analysis of symmetric and asymmetric key lengths. RSA Laboratories' Bulletin, (revised 2001) (2000), http://www.rsasecurity.com/rsalabs/node.asp?id=2088.

Google Scholar

[4]

Leyland, P., Lenstra, A., Dodson, B., Muffett, A., Wagstaff, S.: MPQS with three large primes. In: Fieker, C., Kohel, D.R. (eds.) ANTS-V. LNCS, vol. 2369, pp. 446-460. Springer, Heidelberg (2002).

Digital Library

Google Scholar

[5]

Aoki, K., Kida, Y., Shimoyama, T., Ueda, H.: GNFS176 (2005), http://www.crypto-world.com/announcements/c176.txt

Google Scholar

[6]

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).

Digital Library

Google Scholar

[7]

Coppersmith, D.: Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm. Mathematics of Computation 62, 333-350 (2004).

Digital Library

Google Scholar

[8]

Montgomery, P.L.: Distributed linear algebra. In: presentation slides for ECC2000 (2000) http://www.cacr.math.uwaterloo.ca/conferences/2000/ecc2000/ montgomery.ps

Google Scholar

[9]

Golliver, R.A., Lenstra, A.K., McCurley, K.S.: Lattice sieving and trial division. In: Huang, M-D.A., Adleman, L.M. (eds.) ANTS-I. LNCS, vol. 877, pp. 18-27. Springer, Heidelberg (1994).

Digital Library

Google Scholar

[10]

Aoki, K., Kida, Y., Shimoyama, T., Sonoda, Y., Ueda, H.: SNFS248 (2004), http://www.rkmath.rikkyo.ac.jp/~kida/snfs248e.htm

Google Scholar

Cited By

View all

Zhou TJiang J(2016)Performance modeling of hyper-scale custom machine for the principal steps in block Wiedemann algorithmThe Journal of Supercomputing10.1007/s11227-016-1767-y72:11(4181-4203)Online publication date: 1-Nov-2016
https://dl.acm.org/doi/10.1007/s11227-016-1767-y
Hayashi TShimoyama TShinohara NTakagi T(2012)Breaking pairing-based cryptosystems using ηT pairing over GF(397)Proceedings of the 18th international conference on The Theory and Application of Cryptology and Information Security10.1007/978-3-642-34961-4_5(43-60)Online publication date: 2-Dec-2012
https://dl.acm.org/doi/10.1007/978-3-642-34961-4_5
Shinohara NShimoyama THayashi TTakagi T(2012)Key length estimation of pairing-based cryptosystems using ηT pairingProceedings of the 8th international conference on Information Security Practice and Experience10.1007/978-3-642-29101-2_16(228-244)Online publication date: 9-Apr-2012
https://dl.acm.org/doi/10.1007/978-3-642-29101-2_16
Show More Cited By

Recommendations

Extended Tower Number Field Sieve: A New Complexity for the Medium Prime Case
Proceedings, Part I, of the 36th Annual International Cryptology Conference on Advances in Cryptology --- CRYPTO 2016 - Volume 9814

We introduce a new variant of the number field sieve algorithm for discrete logarithms in $$\mathbb {F}_{p^n}$$ called exTNFS. The most important modification is done in the polynomial selection step, which determines the cost of the whole algorithm: if ...
An experiment of number field sieve over GF(p) of low hamming weight characteristic
IWCC'11: Proceedings of the Third international conference on Coding and cryptology

The security of the digital signature algorithm (DSA) and Diffie-Hellman key exchange is based on the difficulty of the discrete logarithm problems (DLP) over prime field GF(p), and thus it is important to evaluate the difficulty of the DLP over GF(p) ...
Discrete logarithms in GF(P) using the number field sieve

Comments

Information & Contributors

Information

Published In

IWSEC'07: Proceedings of the Security 2nd international conference on Advances in information and computer security

October 2007

460 pages

ISBN:3540756507

Editors:
Atsuko Miyaji
Japan Advanced Institute of Science and Technology, School of Information Science, Ishikawa, Japan
,
Hiroaki Kikuchi
Tokai University, School of Information Technology and Electronics, Kanagawa, Japan
,
Kai Rannenberg
Goethe University, Frankfurt Institute of Business Informatics, Frankfurt/Main, Germany

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 29 October 2007

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
1
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 26 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Zhou TJiang J(2016)Performance modeling of hyper-scale custom machine for the principal steps in block Wiedemann algorithmThe Journal of Supercomputing10.1007/s11227-016-1767-y72:11(4181-4203)Online publication date: 1-Nov-2016
https://dl.acm.org/doi/10.1007/s11227-016-1767-y
Hayashi TShimoyama TShinohara NTakagi T(2012)Breaking pairing-based cryptosystems using ηT pairing over GF(397)Proceedings of the 18th international conference on The Theory and Application of Cryptology and Information Security10.1007/978-3-642-34961-4_5(43-60)Online publication date: 2-Dec-2012
https://dl.acm.org/doi/10.1007/978-3-642-34961-4_5
Shinohara NShimoyama THayashi TTakagi T(2012)Key length estimation of pairing-based cryptosystems using ηT pairingProceedings of the 8th international conference on Information Security Practice and Experience10.1007/978-3-642-29101-2_16(228-244)Online publication date: 9-Apr-2012
https://dl.acm.org/doi/10.1007/978-3-642-29101-2_16
Schmidt BAribowo HDang H(2011)Iterative sparse Matrix-Vector multiplication for integer factorization on GPUsProceedings of the 17th international conference on Parallel processing - Volume Part II10.5555/2033408.2033457(413-424)Online publication date: 29-Aug-2011
https://dl.acm.org/doi/10.5555/2033408.2033457

Cited By

Recommendations

Extended Tower Number Field Sieve: A New Complexity for the Medium Prime Case

An experiment of number field sieve over GF(p) of low hamming weight characteristic

Discrete logarithms in GF(P) using the number field sieve

Comments

Published In

Sponsors

Publisher

Publication History

Author Tags

Qualifiers

Other Metrics

Article Metrics

Other Metrics

Cited By

Abstract

References

Cited By

Recommendations

Extended Tower Number Field Sieve: A New Complexity for the Medium Prime Case

An experiment of number field sieve over GF(p) of low hamming weight characteristic

Discrete logarithms in GF(P) using the number field sieve

Comments

Information

Published In

Sponsors

Publisher

Publication History

Author Tags

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

Share

Share this Publication link

Share on social media

Affiliations