Fürer's algorithm

Fürer's algorithm is an integer multiplication algorithm for quite large integers possessing a very low asymptotic complexity, which can be optimized by using the inverse Ackermann function instead of the iterated logarithm. It was designed in 2007 by the Swiss mathematician Martin Fürer of Pennsylvania State University as an asymptotically faster algorithm (when analysed on a multitape Turing machine) than its predecessor, the Schönhage-Strassen algorithm published in 1971.

The predecessor to Fürer's algorithm, the Schönhage-Strassen algorithm, used Fast Fourier Transform (FFT) to compute integer products in time ${\displaystyle O(n\log n\log \log n)}$ (in big O notation) and its authors, Arnold Schönhage and Volker Strassen, also conjectured a lower bound for the problem of ${\displaystyle \Omega (n\log n)}$. Here ${\displaystyle n}$ 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 ${\displaystyle n}$ in time ${\displaystyle n\log n\,2^{O(\log ^{*}n)}}$ (or by a circuit with that many logic gates), where log*n stands for the iterated logarithm. The difference between the terms ${\displaystyle \log \log n}$ and ${\displaystyle 2^{\log ^{*}n}}$ from a complexity point of view is asymptotically in the advantage of Fürer's algorithm but its asymptotic approximation is only efficient for great numbers. However the difference between the ${\displaystyle \log \log n}$ and ${\displaystyle 2^{\log ^{*}n}}$ factors in the time bounds of Schönhage-Strassen's algorithm and Fürer's algorithm for realistic values of ${\displaystyle n}$ is very small.

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