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Note added June 10, 2009: Part II of this work was deliberate to deal with integer entries, but David Harvey was faster than us, see The Karatsuba middle product for integers. In the following part, we'll dig deep and find out about an archeologist's work. Following work from Cheng, Gergel, Kim and Zima, we current a brand new algorithm with the identical asymptotic complexity, however extra efficient in practice. Then, we present that this second problem could be solved effectively by extending Coppersmith's work on the integer small worth problem--for polynomials with integer coefficients--utilizing lattice discount. Note added on February 10, 2020: in the IEEE TC version, web page 344, Section 5,1, line 6 of the matrix: in column 1, the coefficient should learn Ca instead of C; the coefficient Cb should be in the second column, and the coefficient Cc ought to be within the third column. 2006: Asymptotically Fast Division for GMP, October 2005, revised August 2006, October 2006 and February 2015. - Until model 4.2.1, GNU MP (GMP for brief) division has complexity O(M(n) log(n)), which is not asymptotically optimum

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Searching Worst Cases of a One-Variable Function Using Lattice Reduction, with Damien Stehlé and Vincent Lefèvre, IEEE Transactions on Computers, quantity 54, quantity 3, pages 340-346, 2005. A preliminary version appeared as INRIA Research Report 4586. Some results for the 2x perform double-prolonged precision are available right here. Techniques algorithmiques et méthodes de programmation (in french), 11 pages, July 2005, appeared in Encyclopédie de l'informatique et des systèmes d'info, pages 929-935, Vuibert, 2006. 5,341,321, June 2005. - This brief note exhibits the nasty results of patents for the event of free software, even for patents that weren't written with software program applications in mind. The Elliptic Curve Method, November 2002, revised April 2003 and September 2010, appeared within the Encyclopedia of Cryptography and Security, Springer, 2005 (old hyperlink). An extended note on Mulders' brief product, with Guillaume Hanrot, INRIA Research Report RR-4654, November 2002. A revised model appeared within the Journal of Symbolic Computation, quantity 37, pages 391-401, 2004. A corrigendum appeared in 2014 (pdf). 2005: MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding, with Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, INRIA Research Report RR-5753, November 2005. A revised model appeared in ACM TOMS (Transactions on Mathematical Software), volume 33, number 2, article 13, 2007. - This paper presents a multiple-precision binary floating-level library, written in the ISO C language, and based mostly on the GNU MP library

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