Fürer's algorithm
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(nlognloglogn) in big O notation and its authors, Arnold Schönhage and Volker Strassen, also conjectured a lower bound for the problem of Ω(nlog 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 
(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 
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.
See also
References
- ^ 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
- ^ A. Schönhage and V. Strassen, "Schnelle Multiplikation großer Zahlen", Computing 7 (1971), pp. 281–292.
- ^ 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
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