Schnelle Multiplikation großer Zahlen

@article{Schnhage2005SchnelleMG,
  title={Schnelle Multiplikation gro{\ss}er Zahlen},
  author={Arnold Sch{\"o}nhage and Volker Strassen},
  journal={Computing},
  year={2005},
  volume={7},
  pages={281-292}
}
ZusammenfassungEs wird ein Algorithmus zur Berechnung des Produktes von zweiN-stelligen Dualzahlen angegeben. Zwei Arten der Realisierung werden betrachtet: Turingmaschinen mit mehreren Bändern und logische Netze (aus zweistelligen logischen Elementen aufgebaut).SummaryAn algorithm is given for computing the product of twoN-digit binary numbers byO (N lgN lg lgN) steps. Two ways of implementing the algorithm are considered: multitape Turing machines and logical nets (with step=binary logical… 
Algorithmen – Berechenbarkeit und Komplexität
In den vorigen Kapiteln wurde gezeigt, dass die durch einen Computer zu bearbeitenden Aufgaben durch eine endliche Folge elementarer Anweisungen beschrieben werden mussen, und zwar letztlich in
Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2
TLDR
Polynomial multiplication of degree N can be accomplished in time O (N · log N) provided the scalar field contains suitable roots of unity and the Schönhage-Strassen multiplication is employed.
Fast integer multiplication using \goodbreak generalized Fermat primes
TLDR
An alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, is used to obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model.
Über die Komplexität der Multiplikation in eingeschränkten Branchingprogrammmodellen
TLDR
Theoretisch schnellste bekannte Verfahren zur Multiplikation n-stelliger ganzer Zahlen stammt von von Schönhage and Strassen [SS71] and hat einen Zeitbedarf von O(n log n log log log n).
Reversible Karatsuba's Algorithm
Karatsuba discovered the first algorithm that accomplishes multiprecision integer multiplication with complexity below that of the grade-school method. This al- gorithm is implemented nowadays in
Wie kann man Primzahlen erkennen
Dass die Aufgabe, die Primzahlen von den zusammengesetzten zu unterscheiden und letztere in ihre Primfactoren zu zerlegen, zu den wichtigsten und nutzlichsten der gesamten Arithmetik gehort und die
Fast Polynomial Multiplication over F260
TLDR
This paper shows how central ideas of the recent asymptotically fast algorithms turn out to be of practical interest for multiplication of polynomials over finite fields of characteristic two, and presents the Mathemagix implementation, which outperforms existing implementations in large degree.
Fast arithmetic for faster integer multiplication
TLDR
This work obtains the same result K = 4 using simple modular arithmetic as a building block, and a careful complexity analysis, based on a conjecture about the existence of sufficiently many primes of a certain form.
A gmp-based implementation of schönhage-strassen's large integer multiplication algorithm
TLDR
An improved implementation of Schönhage-Strassen's algorithm, based on the one distributed within the GMP library is presented, with improvements to faster arithmetic modulo 2n + 1, improved cache locality and tuning.
Fast integer multiplication using generalized Fermat primes
TLDR
An alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, is used to obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model.
...
...

References

SHOWING 1-10 OF 16 REFERENCES
Multiplikation großer Zahlen
TLDR
The translations needed between the binary form of numbers and their representation by residues can be performed sufficiently fast by use of a modul of the form Π (2q i–1) by a recursive iteration of this principle and additional numbertheoretical arguments.
An algorithm for the machine calculation of complex Fourier series
TLDR
Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
The Art of Computer Programming
TLDR
The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Die Komplexi t~t eines logischen Netzes, das die Multiplikation ganzer Zahlen realisiert
  • Dokl. Akad. Nauk SSSR 150,
  • 1963
Insgesamt haben wir gezeigt, dab sich die Multiplikation in ZF2,_ ~ mit 2 n Multiplikationen in ZF n und zus/~tzlichem Aufwand 0 (n-22n) realisieren l~fit
    TU~:Eu An Algorithm for the Machine CMculatiom of
    • Complex FouRIEa Series. Math. Comp
    • 1965
    Multiplikation vielstelliger Zahlen mi t Reehen - automaten ( russisch )
    • Dokl . Akad .
    • 1965
    AA ~ DERAA : On the Minimum Computat ion Time of Functions
    • Trans . AMS
    • 1966
    AA~DERAA: On the Minimum Computat ion
    • Time of Functions. Trans. AMS 142~
    • 1969
    ...
    ...