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Integer and polynomial multiplication: towards optimal toom-cook matrices

Published: 29 July 2007 Publication History

Abstract

Karatsuba and Toom-Cook are well-known methods used to multiply efficiently long integers. There have been different proposal about the interpolating values used to determine the matrix to be inverted and the sequence of operations to invert it. A deffinitive word about which is the optimal matrix (values) and the (number of) basic operations to invert it seems still not to have been said. In this paper we present some particular examples of useful matrices and a method to generate automatically, by means of optimised exhaustive searches on a graph, the best sequence of basic operations to invert them.

References

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M. Bodrato. Towards optimal Toom-Cook multiplication for univariate and multivariate polynomials in characteristic 2 and 0. In C. Carlet and B. Sunar, editors, WAIFI 2007 proceedings, volume 4547 of LNCS, pages 116--133. Springer, June 2007. http://bodrato.it/papers/#WAIFI2007.
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M. Bodrato and A. Zanoni. What about Toom-Cook matrices optimality ? Technical Report 605, Centro "Vito Volterra", Università di Roma "Tor Vergata", October 2006. http://bodrato.it/papers/#CIVV2006.
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J. Chung and M. A. Hasan. Asymmetric squaring formulae. Technical Report 24, University of Waterloo, August 2006.
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J. Chung and M. A. Hasan. Asymmetric squaring formulae. In Proceedings of the ARITH-18 conference. IEEE, June 2007.
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S. A. Cook. On the minimum computation time of functions. PhD thesis, Dept. of Mathematics, Harvard University, 1966.
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P. Gaudry, A. Kruppa, and P. Zimmermann. A GMP-based implementation of Schönhage-Strassen's large integer multiplication algorithm. In C. W. Brown, editor, Proceedings of the ISSAC 2007 conference. ACM press, July 2007.
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GNU MP: The GNU multiple precision arithmetic library, v4.2.1, 2006. http://gmplib.org/#DOC.
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A. A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Soviet Physics Doklady, 7(7):595--596, 1963.
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cover image ACM Conferences
ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
July 2007
406 pages
ISBN:9781595937438
DOI:10.1145/1277548
  • General Chair:
  • Dongming Wang
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 29 July 2007

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Author Tags

  1. Karatsuba
  2. integer and polynomial multiplication
  3. interpolation
  4. matrix inversion
  5. squaring
  6. toom-cook

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ISSAC07
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ISSAC07: International Symposium on Symbolic and Algebraic Computation
July 29 - August 1, 2007
Ontario, Waterloo, Canada

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Overall Acceptance Rate 395 of 838 submissions, 47%

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  • (2024)GCKSign: Simple and efficient signatures from generalized compact knapsack problemsPLOS ONE10.1371/journal.pone.031070819:9(e0310708)Online publication date: 23-Sep-2024
  • (2024)Fault-Tolerant Parallel Integer MultiplicationProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659961(207-218)Online publication date: 17-Jun-2024
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  • (2022)A strategy to optimize the complexity of Chudnovsky-type algorithms over the projective lineArithmetic, Geometry, Cryptography, and Coding Theory 202110.1090/conm/779/15668(13-32)Online publication date: 2022
  • (2021)Fast NEON-Based Multiplication for Lattice-Based NIST Post-quantum Cryptography FinalistsPost-Quantum Cryptography10.1007/978-3-030-81293-5_13(234-254)Online publication date: 20-Jul-2021
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