Fürer's algorithm

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Fürer's algorithm is an integer multiplication algorithm for extremely large integers with very low asymptotic complexity. It was published in 2007 by the Swiss mathematician Martin Fürer of Pennsylvania State University[1] as an asymptotically faster algorithm when analysed on a multitape Turing machine than its predecessor, the Schönhage–Strassen algorithm.[2]

The Schönhage–Strassen algorithm uses the fast Fourier transform (FFT) to compute integer products in time and its authors, Arnold Schönhage and Volker Strassen, conjecture a lower bound of . Fürer's algorithm reduces the gap between these two bounds. It can be used to multiply integers of length in time where log*n is the iterated logarithm. The difference between the and terms, from a complexity point of view, is asymptotically in the advantage of Fürer's algorithm for integers greater than . However the difference between these terms for realistic values of is very small.[1]

In 2008 De et al gave a similar algorithm which relies on modular arithmetic instead of complex arithmetic to achieve in fact the same running time.[3] It has been estimated that it becomes faster than Schönhage-Strassen for inputs exceeding a length of .[4]

In 2015 Harvey, van der Hoeven and Lecerf[5] gave a new algorithm that achieves a running time of making explicit the implied constant in the exponent. They also proposed a variant of their algorithm which achieves but whose validity relies on standard conjectures about the distribution of Mersenne primes.

In 2015 Covanov and Thomé provided another modification of Fürer's algorithm which achieves the same running time.[6] Nevertheless, it remains just as impractical as all the other implementations of this algorithm.[7][8]

In 2016 Covanov and Thomé proposed an integer multiplication algorithm based on a generalization of Fermat primes that conjecturally achieves a complexity bound of . This matches the 2015 conditional result of Harvey, van der Hoeven, and Lecerf but uses a different algorithm and relies on a different conjecture.[9]

In 2018 Harvey and van der Hoeven used an approach based on the existence of short lattice vectors guaranteed by Minkowski's theorem to prove an unconditional complexity bound of .[10]

See also[edit]

References[edit]

  1. ^ a b M. Fürer (2007). "Faster Integer Multiplication" Proceedings of the 39th annual ACM Symposium on Theory of Computing (STOC), 55–67, San Diego, CA, June 11–13, 2007, and SIAM Journal on Computing, Vol. 39 Issue 3, 979–1005, 2009.
  2. ^ A. Schönhage and V. Strassen (1971). "Schnelle Multiplikation großer Zahlen", Computing, Vol. 7, 281–292, 1971.
  3. ^ A. De, C. Saha, P. Kurur and R. Saptharishi (2008). "Fast integer multiplication using modular arithmetic" Proceedings of the 40th annual ACM Symposium on Theory of Computing (STOC), 499–506, New York, NY, 2008, and SIAM Journal on Computing, Vol. 42 Issue 2, 685–699, 2013. arXiv:0801.1416
  4. ^ Lüders, Christoph (2015). Implementation of the DKSS Algorithm for Multiplication of Large Numbers (pdf). International Symposium on Symbolic and Algebraic Computation. pp. 267–274. 
  5. ^ D. Harvey, J. van der Hoeven and G. Lecerf (2015). "Even faster integer multiplication", February 2015. arXiv:1407.3360
  6. ^ S. Covanov and E. Thomé (2015). "Fast arithmetic for faster integer multiplication", 2015.
  7. ^ S. Covanov and E. Thomé (2014). "Efficient implementation of an algorithm multiplying big numbers", Internal research report, Polytechnics School, France, July 2014.
  8. ^ S. Covanov and M. Moreno Mazza (2015). "Putting Fürer algorithm into practice", Master research report, Polytechnics School, France, January 2015.
  9. ^ S. Covanov and E. Thomé (2016). "Fast integer multiplication using generalized Fermat primes", paper submitted to the Journal on Mathematics of Computation, January 2016. arXiv:1502.02800
  10. ^ D. Harvey and J. van der Hoeven (2018). "Faster integer multiplication using short lattice vectors", February 2018. arXiv:1802.07932