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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 [edit]

^ ^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

v t e Number-theoretic algorithms

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

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 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 Integer relation Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms Smallcaps indicate a deterministic algorithm

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Fürer's algorithm

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