Multiplication algorithm: Difference between revisions

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{{unsolved|computer science|What is the fastest algorithm for multiplication of two <math>n</math>-digit numbers?}}

A line of research in [[theoretical computer science]] is about the number of single-bit arithmetic operations necessary to multiply two <math>n</math>-bit integers. This is known as the [[computational complexity]] of matrix multiplication. Usual algorithms done by hand have asymptotic complexity of <math>O(n^2)</math>, but in 1960 [[Anatoly Karatsuba]] discovered that better complexity was possible (with the [[Karatsuba algorithm]]).

The algorithm with the best computational complexity is a 2019 algorithm of [[David Harvey (mathematician)|David Harvey]] and [[Joris van der Hoeven]], which uses the strategies of using [[number-theoretic transform]]s introduced with the [[Schönhage–Strassen algorithm]] to multiply integers using only <math>O(n\log n)</math> operations.<ref>{{cite journal | last1 = Harvey | first1 = David | last2 = van der Hoeven | first2 = Joris | author2-link = Joris van der Hoeven | doi = 10.4007/annals.2021.193.2.4 | issue = 2 | journal = [[Annals of Mathematics]] | mr = 4224716 | pages = 563–617 | series = Second Series | title = Integer multiplication in time <math>O(n \log n)</math> | volume = 193 | year = 2021}}</ref> This is conjectured to be the best possible algorithm, but lower bounds of <math>\Omega(n\log n)</math> are not known.