Fürer's algorithm

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Fürer's algorithm is an integer multiplication algorithm for very large numbers possessing a very low asymptotic complexity. It was created in 2007 by Swiss mathematician Martin Fürer of Pennsylvania State University[1] as an asymptotically faster (when analysed on a multitape Turing machine) algorithm than its predecessor, the Schönhage–Strassen algorithm published in 1971.[2]

The predecessor to the Fürer algorithm, the Schönhage–Strassen algorithm, used fast Fourier transforms to compute integer products in time O(n \log n \log \log n) (in big O notation) and its authors, Arnold Schönhage and Volker Strassen, also conjectured a lower bound for the problem of \Omega(n \log n). Here, n denotes the total number of bits in the two input numbers. Fürer's algorithm reduces the gap between these two bounds: it can be used to multiply binary integers of length n in time n \log n\,2^{O(\log^* n)} (or by a circuit with that many logic gates), where log*n represents the iterated logarithm operation. However, the difference between the \log \log n and 2^{\log^* n} factors in the time bounds of the Schönhage–Strassen algorithm and the Fürer algorithm for realistic values of n is very small.[1]

In 2008, Anindya De, Chandan Saha, Piyush Kurur and Ramprasad Saptharishi[3] gave a similar algorithm that relies on modular arithmetic instead of complex arithmetic to achieve the same running time.

In 2014, David Harvey, Joris van der Hoeven, and Grégoire Lecerf[4] showed that an optimized version of Fürer's algorithm achieves a running time of O(n\log n2^{4\log^* n}), making the implied constant in the O(\log^* n) exponent explicit. They also gave a modification to Fürer's algorithm that achieves O(n\log n2^{3\log^* n})

In 2015 Svyatoslav Covanov and Emmanuel Thomé provided another modifications that achieves same running time.[5] Yet, as all the other implementation, it's still not practical.[citation needed]

See also[edit]


  1. ^ a b Fürer, M. (2007). "Faster Integer Multiplication" in Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11–13, 2007, San Diego, California, USA
  2. ^ A. Schönhage and V. Strassen, "Schnelle Multiplikation großer Zahlen", Computing 7 (1971), pp. 281–292.
  3. ^ Anindya De, Piyush P Kurur, Chandan Saha, Ramprasad Saptharishi. Fast Integer Multiplication Using Modular Arithmetic. Symposium on Theory of Computation (STOC) 2008. arXiv:0801.1416
  4. ^ David Harvey, Joris van der Hoeven, and Grégoire Lecerf, "Even faster integer multiplication", 2014, arXiv:1407.3360
  5. ^ Svyatoslav Covanov and Emmanuel Thomé, "Fast arithmetic for faster integer multiplication", 2015 arXiv:1502.02800