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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.

## References

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

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Fürer's_algorithm — Please support Wikipedia.
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