# Fürer's algorithm

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

## Background

The Schönhage–Strassen algorithm uses the fast Fourier transform (FFT) to compute integer products in time ${\displaystyle O(n\log n\cdot \log \log n)}$  and its authors, Arnold Schönhage and Volker Strassen, conjecture a lower bound of ${\displaystyle \Omega (n\log n)}$ . Fürer's algorithm reduces the gap between these two bounds. It can be used to multiply integers of length ${\displaystyle n}$  in time ${\displaystyle O\left(n\log n\ \cdot 2^{O(\log ^{*}n)}\right)}$  where log*n is the iterated logarithm. The difference between the ${\displaystyle \log \log n}$  and ${\displaystyle 2^{\log ^{*}n}}$  terms, from a complexity point of view, is asymptotically in the advantage of Fürer's algorithm for integers greater than ${\displaystyle 2^{2^{64}}}$ . However the difference between these terms for realistic values of ${\displaystyle n}$  is very small.[1]

## Improved algorithms

In 2008 De et al gave a similar algorithm which relies on modular arithmetic instead of complex arithmetic to achieve in fact the same running time.[3] It has been estimated that it becomes faster than Schönhage-Strassen for inputs exceeding a length of ${\displaystyle 10^{10^{4796}}}$ .[4]

In 2015 Harvey, van der Hoeven and Lecerf[5] gave a new algorithm that achieves a running time of ${\displaystyle O(n\log n\cdot 2^{3\log ^{*}n})}$  making explicit the implied constant in the ${\displaystyle O(\log ^{*}n)}$  exponent. They also proposed a variant of their algorithm which achieves ${\displaystyle O(n\log n\cdot 2^{2\log ^{*}n})}$  but whose validity relies on standard conjectures about the distribution of Mersenne primes.

In 2015 Covanov and Thomé provided another modification of Fürer's algorithm which achieves the same running time.[6] Nevertheless, it remains just as impractical as all the other implementations of this algorithm.[7][8]

In 2016 Covanov and Thomé proposed an integer multiplication algorithm based on a generalization of Fermat primes that conjecturally achieves a complexity bound of ${\displaystyle O(n\log n\cdot 2^{2\log ^{*}n})}$ . This matches the 2015 conditional result of Harvey, van der Hoeven, and Lecerf but uses a different algorithm and relies on a different conjecture.[9]

In 2018 Harvey and van der Hoeven used an approach based on the existence of short lattice vectors guaranteed by Minkowski's theorem to prove an unconditional complexity bound of ${\displaystyle O(n\log n\cdot 2^{2\log ^{*}n})}$ .[10]

In March 2019, Harvey and van der Hoeven published a long-sought after, asymptotically optimal ${\displaystyle O(n\log n)}$  integer multiplication algorithm.[11][12] Because Schönhage and Strassen predicted that n * log(n) is the ‘best possible’ result Harvey said: “...our work is expected to be the end of the road for this problem, although we don't know yet how to prove this rigorously.”[13]

## References

1. ^ a b M. Fürer (2007). "Faster Integer Multiplication" Proceedings of the 39th annual ACM Symposium on Theory of Computing (STOC), 55–67, San Diego, CA, June 11–13, 2007, and SIAM Journal on Computing, Vol. 39 Issue 3, 979–1005, 2009.
2. ^ Schönhage, A.; Strassen, V. (1971). "Schnelle Multiplikation großer Zahlen" [Fast Multiplication of Large Numbers]. Computing (in German). 7 (3–4): 281–292. doi:10.1007/BF02242355.
3. ^ A. De, C. Saha, P. Kurur and R. Saptharishi (2008). "Fast integer multiplication using modular arithmetic" Proceedings of the 40th annual ACM Symposium on Theory of Computing (STOC), 499–506, New York, NY, 2008, and SIAM Journal on Computing, Vol. 42 Issue 2, 685–699, 2013. arXiv:0801.1416
4. ^ Lüders, Christoph (2015). Implementation of the DKSS Algorithm for Multiplication of Large Numbers (pdf). International Symposium on Symbolic and Algebraic Computation. pp. 267–274.
5. ^ D. Harvey, J. van der Hoeven and G. Lecerf (2015). "Even faster integer multiplication", February 2015. arXiv:1407.3360
6. ^ Covanov, S.; Thomé, E. (2015). "Fast Arithmetic for Faster Integer Multiplication". arXiv:1502.02800v1. Cite journal requires |journal= (help) Published as Covanov & Thomé (2019).
7. ^ S. Covanov and E. Thomé (2014). "Efficient implementation of an algorithm multiplying big numbers", Internal research report, Polytechnics School, France, July 2014.
8. ^ S. Covanov and M. Moreno Mazza (2015). "Putting Fürer algorithm into practice", Master research report, Polytechnics School, France, January 2015.
9. ^ Covanov, Svyatoslav; Thomé, Emmanuel (2019). "Fast Integer Multiplication Using Generalized Fermat Primes". Math. Comp. 88: 1449–1477. arXiv:1502.02800. doi:10.1090/mcom/3367.
10. ^ D. Harvey and J. van der Hoeven (2018). "Faster integer multiplication using short lattice vectors", February 2018. arXiv:1802.07932
11. ^ Harvey, David; Van Der Hoeven, Joris (2019-04-12). Integer multiplication in time O(n log n).
12. ^ Hartnett, Kevin (2019-04-14). "Mathematicians Discover the Perfect Way to Multiply". Wired. ISSN 1059-1028. Retrieved 2019-04-15.
13. ^ Gilbert, Lachlan (4 April 2019). "Maths whiz solves 48-year-old multiplication problem". UNSW. Retrieved 18 April 2019.