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 Martin Fürer of Pennsylvania State University^[1] as an asymptotically less-complex 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 $O (n log n loglog n)$ , and its authors, Arnold Schönhage and Volker Strassen, also conjectured a lower bound for the problem to be solved by an unknown algorithm as $O (n log n)$ . 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 Boolean circuit with that many logic gates), where $log * n$ represents the iterated logarithm operation. The difference between the $loglog n$ and $2^{\log^* n}$ factors in the time bounds of the Schönhage-Strassen algorithm and the Fürer algorithm are so small that Fürer's algorithm would only speed up multiplication operations on "astronomically large numbers".^[1]

[edit] See also

Number-theoretic transform

[edit] References

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^ ^a ^b Fuhrer, 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
^ A. Schönhage and V. Strassen, "Schnelle Multiplikation großer Zahlen", Computing 7 (1971), pp. 281–292.