Abstract
The strength of many security protocols lies on the computational intractability of the integer factorization and discrete logarithm problems. Currently, the best-known techniques employed are number field sieve (NFS) family of algorithms. They come under the class of sub-exponential time algorithms. This class of algorithms comprises of multiple steps. The relation collection (sieving step) is one of the computationally costly and highly memory-dependent phase of these algorithms. This paper discusses various ways to improve the efficiency of the relation collection phase by using parallelization techniques. Experiments have been carried out by using function field sieve, which is one of the NFS family algorithms, to show the computation efficiency of parallelization techniques along with the suitable sieving techniques and the key parameters. The result of our basic implementation is compared with the parallelized version of it. The result analysis depicts that the relation collection phase can be improved by using parallelization techniques up to fourfold.
Similar content being viewed by others
References
Rivest R, Shamir A, Adleman L (1978) A method for obtaining digital signatures and public key cryptosystems. Commun ACM 21:120–126
Kleinjung T, Aoki K, Franke J, Lenstra AK, Thome E, Bos J, Gaudry P, Kruppa A, Montgomery PL, Osvik DA, Te Riele H, Timofeev A, Zimmermann P (2010) Factorization of a 768-bit RSA modulus. In: Rabin T (ed) Advances in cryptology: CRYPTO 2010, LNCS, vol 6223, Springer, pp 333–350
Diffie W, Hellman ME (1976) New directions in cryptography. IEEE Trans Inf Theory 22(6):644–654
Barbulescu R, Bouvier C, Detrey J, Gaudry P, Jeljeli H, Thome E, Videau M, Zimmermann P (2014) Discrete logarithm in \(GF(2^{809})\) with FFS, PKC 2014, LNCS, vol 8383, Springer, pp 221–238
https://prog.world/a-new-achievement-in-cryptography-factorization-of-a-795-bit-rsa-number (2019)
Abhijit DA (2013) Computational number theory. Chapman and Hall/CRC 2013 Print. ISBN: 978-1-4398-6615-3
Joux A (2013) A new index calculus algorithm with complexity L(1/4 + o(1)) in very small characteristic. Sel Areas Cryptogr LNCS 8282:355–379
Barbulescu R, Gaudry P, Joux A, Thome E (2013) A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic (preprint), 8 pages. http://hal.inria.fr/hal-00835446
Lenstra AK, Lenstra Jr HW (1993) The development of the number field sieve. Springer, LNM, p 1554
Adleman LM (1994) The function field sieve. In: Algorithmic number theory (ANTS-I), LNCS, vol 877, Spinger, Berlin, pp 108–121
Schirokauer O (2000) Using number fields to compute logarithms in finite fields. Math Comput 69(231):1267–1283
Barbulescu R, Gaudry P, Kleinjung T (2015) The tower number field sieve. In: Advances in cryptology: ASIACRYPT 2015, LNCS, vol 9453, Springer, Berlin
Kim T, Barbulescu R (2016) Extended tower number field sieve: a new complexity for the medium prime case. In: Advances in cryptology—CRYPTO 2016, LNCS, vol 9814, Springer, Berlin
Zhu Y, Wen J, Zhuang J, Lv C, Lin D (2020) Theoretical computer science, vol 814, pp 49–68
Granger R, Kleinjung T, Zumbrgel J (2014) Discrete logarithms in GF (29234)—NMBRTHRY list
Granger R, Kleinjung T, Zumbrgel J (2014) Breaking 128-bit secure supersingular binary curves (or how to solve discrete logarithms in F (24–1223) and F (212–367). In: CRYPTO, vol 17, pp 126–145
Glaglu F, Granger R, McGuire G, Zumbrgel J (2013) On the function field sieve and the impact of higher splitting probabilities: application to discrete logarithms in F (21971) and F (23164). In: Advances in cryptology: CRYPTO 2013, LNCS , vol 8043, pp 109–128
Sarkar Palash (2016) Fine tuning the function field sieve algorithm for the medium prime case. IEEE Trans Inf Theory 62(4):2233–2253
Sarkar P, Singh S (2016) New complexity trade-offs for the (multiple) number field sieve algorithm in nonprime fields. In: Fischlin M, Coron JS (eds) EUROCRYPT 2016. LNCS, vol 9665, Springer, pp 429–458
Kleinjung T (2006) On polynomial selection for the general number field sieve. Math. Comput. 75:2037–2047
Franke J, Kleinjung T (2005) Continued fractions and lattice sieving. In: Proceedings SHARCS 2005. http://www.ruhrunibochum.de/itsc/tanja/SHARCS/talks/FrankeKleinjung.pdf
Aoki K, Ueda H (2004) Sieving using bucket sort. In: Lee PJ (ed) Advances in cryptology: ASIACRYPT 2004. ASIACRYPT, LNCS, vol 3329, Springer, Berlin
Gaudry P, Gremy L, Videau M (2016) Collecting relations for the number field sieve in \(\text{GF}(p^{6})\). LMS J Comput Math 19(A): 332–350. 10.1112/S1461157016000164
Sengupta B, Das Abhijit (2017) Use of SIMD-based data parallelism to speed up aieving in integer-factoring alogrithm. Appl Math Comput 293:204–217
Pollard JM (993) The lattice sieve. Lenstra AK, Lenstra HW Jr (eds) The development of the number field sieve, LNM, vol 1554, Springer, Berlin, pp 43–49
Joux A, Lercier R (2001) Discrete logarithms in \(GF(2^n )\) (521 bits), email to the NMBRTHRY mailing list. http://listserv.nodak.edu/archives/nmbrthry.html
Joux A, Lercier R (2005) Discrete logarithms in \(GF(2^{607})\) and \(GF(2^{613})\). E-mail to the NMBRTHRY mailing list. http://listserv.nodak.edu/archives/nmbrthry
Herstein IN (1975) Topics in algebra, 2nd edn. ISBN:978-0-471-01090-6
Jarvis F (2014) Algebraic number theory. Springer, Berlin
Joux A (2009) Algorithmic cryptanalysis, Chapman and Hall/CRC 2009 Print ISBN: 978-1-4200-7002-6 eBook ISBN: 978-1-4200-7003-3. https://doi.org/10.1201/9781420070033
Case M (2003) A beginners guide to the general number field sieve. http://islab.oregonstate.edu/koc/ece575/03Project/Case/paper.pdf
Joux A, Lercier R (2002) The function field sieve is quite special. In: Algorithmic numberv theory-ANTS V, LNCS, vol 2369, Springer, pp 431–445
Barbulescu R (2013) Selecting polynomials for the function field sieve, preprint, p 23. http://hal.inria.fr/hal-00798386
Golliver RA, Lenstra AK, McCurley KS (1994) Lattice sieving and trial division. In: Adleman LM, Huang MD (eds) Algorithmic number theory, LNCS, vol 877, Springer, pp 18–27
JDetrey J, Gaudry P, Videau M (2013) Relation collection for the function field sieve. In: Nannarelli A, Seidel PM, Tang PTP (eds) Proceedings of ARITH-21, pp 201–210
GGNFS Library. http://gilchrist.ca/jeff/factoring/nfs-beginners-guide.html
CADO-NFS Library. http://cado-nfs.gforge.inria.fr/
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Varshney, S., Charpe, P., Padmavathy, R. et al. Relation collection using Pollard special-q sieving to solve integer factorization and discrete logarithm problem. J Supercomput 77, 2734–2769 (2021). https://doi.org/10.1007/s11227-020-03351-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11227-020-03351-6