Abstract
We review several methods for the square root step of the Number Field Sieve, and present an original one, based on the Chinese Remainder Theorem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brent, R.P.: Multiple-precision zero-finding methods and the complexity of elementary function evaluation. In: Traub, J.F. (ed.) Analytic Computational Complexity, pp. 151–176. Academic Press, New York (1975), http://web.comlab.ox.ac.uk/oucl/work/richard.brent/ftp/rpb028.ps.gz
Brent, R., Zimmermann, P.: Modern Computer Arithmetic. Cambridge Monographs on Applied and Computational Mathematics, vol. 18. Cambridge University Press (2010)
Buhler, J.P., Lenstra, A.K., Pollard, J.M.: 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 Math., vol. 1554, pp. 50–94. Springer (1993)
Cohen, H.: A course in algorithmic algebraic number theory. Grad. Texts in Math., vol. 138. Springer (1993)
Couveignes, J.-M.: Computing a square root for the number field sieve. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The Development of the Number Field Sieve. Lecture Notes in Math., vol. 1554, pp. 95–102. Springer (1993)
Enge, A., Sutherland, A.V.: Class Invariants by the CRT Method. In: Hanrot, G., Morain, F., Thomé, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 142–156. Springer, Heidelberg (2010)
von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (1999)
Gaudry, P., Kruppa, A., Morain, F., Muller, L., Thomé, E., Zimmermann, P.: cado-nfs, An Implementation of the Number Field Sieve Algorithm (2011), http://cado-nfs.gforge.inria.fr/ , Release 1.1
Howgrave-Graham, N., Joux, A.: New Generic Algorithms for Hard Knapsacks. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 235–256. Springer, Heidelberg (2010)
Joux, A., Naccache, D., Thomé, E.: When e-th Roots Become Easier Than Factoring. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 13–28. Springer, Heidelberg (2007)
Kleinjung, T., Aoki, K., Franke, J., Lenstra, A.K., Thomé, E., Bos, J.W., Gaudry, P., Kruppa, A., Montgomery, P.L., Osvik, D.A., te Riele, H., Timofeev, A., Zimmermann, P.: Factorization of a 768-Bit RSA Modulus. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 333–350. Springer, Heidelberg (2010)
Lenstra Jr., H.W., Stevenhagen, P.: Chebotarëv and his density theorem. Math. Intelligencer 18(2), 26–37 (1996)
Monico, C.: ggnfs, A Number Field Sieve Implementation (2004-2005), http://www.math.ttu.edu/~cmonico/software/ggnfs/ , Release 0.77
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. Proc. Sympos. Appl. Math., vol. 48, pp. 567–571. Amer. Math. Soc. (1994)
Montgomery, P.L.: Square roots of products of algebraic numbers (1997), unpublished draft, significantly different from published version [14] (May 16, 1997)
Nguyên, P.Q.: 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)
Papadopoulos, J.: msieve, A Library for Factoring Large Integers – release 1.50 (2004), http://www.boo.net/~jasonp , Release 1.50
Sutherland, A.V.: Accelerating the CM method (2012) (preprint), http://arxiv.org/abs/1009.1082
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Thomé, E. (2012). Square Root Algorithms for the Number Field Sieve. In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-31662-3_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31661-6
Online ISBN: 978-3-642-31662-3
eBook Packages: Computer ScienceComputer Science (R0)