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 ^[1] as an asymptotically faster algorithm (when analysed on a multitape Turing machine) 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 in order 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.

In 2014, David Harvey, Joris van der Hoeven and Grégoire Lecerf ^[4] showed that an optimized version of Fürer's algorithm achieves a running time of $O(n\log n2^{3\log ^{*}n})$ , making the implied constant in the $O(\log ^{*}n)$ exponent explicit. They also gave a modification to Fürer's algorithm that achieves $O(n\log n2^{2\log ^{*}n})$ .

In 2015, Svyatoslav Covanov and Emmanuel Thomé provided another modification that achieves the same running time. ^[5] Yet it remains just as impractical as the other implementations of Fürer's algorithm. ^[6] ^[7]

References

1 2 M. Fürer (2007), "Faster Integer Multiplication" Proceedings of the 39th annual ACM Symposium on Theory of Computing, pp. 55-67, June 11-13, 2007, San Diego, California, USA, and SIAM Journal on Computing, Vol. 39 Issue 3, pp. 979-1005, 2007.
↑ A. Schönhage and V. Strassen (1971), "Schnelle Multiplikation großer Zahlen", Computing 7, pp. 281–292, 1971.
↑ A. De, C. Saha, P. Kurur and R. Saptharishi (2008). "Fast integer multiplication using modular arithmetic" Proceedings of the Symposium on Theory of Computation (STOC), 2008. arXiv:0801.1416
↑ D. Harvey, J. van der Hoeven and G. Lecerf (2014), "Even faster integer multiplication", 2014, arXiv:1407.3360
↑ S. Covanov and E. Thomé (2015), "Fast arithmetic for faster integer multiplication", 2015 arXiv:1502.02800
↑ S. Covanov (2015), "Putting Fürer algorithm into practice", 2015.
↑ S. Covanov and E. Thomé (2016), "Fast integer multiplication using generalized Fermat primes", 2016.

Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia LLL Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms

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Fürer's algorithm

See also

References