Due to its overhead, Toom Cook is slower than long multiplication with small numbers, and it is therefore typically used for intermediate-size multiplications, before the asymptotically faster Sch�nhage Strassen algorithm ( with complexity ) becomes practical.
12.
The Sch�nhage Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007, when a new method, F�rer's algorithm, was announced with lower asymptotic complexity; however, F�rer's algorithm currently only achieves an advantage for astronomically large values and is not used in practice.
13.
As describes, the time for finding the fundamental solution using the continued fraction method, with the aid of the Sch�nhage Strassen algorithm for fast integer multiplication, is within a logarithmic factor of the solution size, the number of digits in the pair ( " x " 1, " y " 1 ).
14.
Applications of the Sch�nhage Strassen algorithm include mathematical empiricism, such as the Great Internet Mersenne Prime Search and computing approximations of " ? ", as well as practical applications such as Kronecker substitution, in which multiplication of polynomials with integer coefficients can be efficiently reduced to large integer multiplication; this is used in practice by GMP-ECM for Lenstra elliptic curve factorization.
15.
The predecessor to F�rer's algorithm, the Sch�nhage-Strassen algorithm, used Fast Fourier Transform ( FFT ) to compute integer products in time O ( n \ log n \ log \ log n ) ( in big O notation ) and its authors, Arnold Sch�nhage and Volker Strassen, also conjectured a lower bound for the problem of Here n denotes the total number of bits in the two input numbers.
16.
For multiplication, the most straightforward algorithms used for multiplying numbers by hand ( as taught in primary school ) require O ( " N " 2 ) operations, but multiplication algorithms that achieve O ( " N " log ( " N " ) log ( log ( " N " ) ) ) complexity have been devised, such as the Sch�nhage Strassen algorithm, based on fast Fourier transforms, and there are also algorithms with slightly worse complexity but with sometimes superior real-world performance for smaller " N ".
