Multiplication

Description

Multiplication is one of the four elementary mathematical operations of arithmetic; with the others being addition; subtraction and division. Given two $n$-bit integers, compute their product, which should be a $2n$-bit integer.

Parameters

$n$: length of one of the integers, in bits

Table of Algorithms

Name	Year	Time	Space	Approximation Factor	Model	Reference
Karatsuba Algorithm	1962	$O(n^{1.{5}8})$	$O(n)$	Exact	Deterministic	Time
Toom-3	1969	$O(n^{1.{4}6})$	$O(n)$	Exact	Deterministic	Time
Long Multiplication	1940	$O(n^{2})$	$O(n)$	Exact	Deterministic
Schönhage–Strassen algorithm	1971	$O(n \log n \log\log n)$	$O(n)$ auxiliary?	Exact	Deterministic	Time
Furer's algorithm	2007	$O(n \log n {2}^{O(\log*n)})$	$O(n)$ auxiliary?	Exact	Deterministic	Time
De	2008	$O(n \log n {2}^{O(\log*n)})$	$O(n)$ auxiliary?	Exact	Deterministic	Time
Harvey; Hoeven	2019	$O(n \log n)$	$O(n)$ auxiliary??	Exact	Deterministic	Time
Harvey; Hoeven; Lecerf	2015	$O(n \log n {2}^{({3} \log*n)})$	$O(n)$ auxiliary??	Exact	Deterministic	Time
Covanov and Thomé	2015	$O(n \log n {2}^{O(\log*n)})$	$O(n)$ auxiliary??	Exact	Deterministic	Time
Covanov and Thomé	2016	$O(n \log n {2}^{({3} \log*n)})$	$O(n)$ auxiliary??	Exact	Deterministic	Time
Harvey; Hoeven; Lecerf	2018	$O(n \log n {2}^{({2} \log*n)})$	$O(n)$ auxiliary??	Exact	Deterministic	Time

Time Complexity Graph

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Space Complexity Graph

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Time-Space Tradeoff

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References/Citation

https://hal.archives-ouvertes.fr/hal-02070778