This page says following:

Integer Multiplication has an O-optimal linear-time algorithm on a RAM or SMM

Is this page fooling me or how can we multiply 2 numbers in linear time (bitwise complexity) using RAM?

  • $\begingroup$ Why do you think the page is talking about bit complexity? $\endgroup$ Sep 9, 2017 at 10:21
  • $\begingroup$ Because when speaking of words it takes $O(1)$ time. $\endgroup$
    – rus9384
    Sep 9, 2017 at 11:05

1 Answer 1


The page is not fooling you. I'll refer you to the original paper, but accordingly they show the following:

Theorem 6.1 There exists an SMM which performs integer-multiplication in linear time, i.e. there is a constant $c$ such that any $2N$-bit input is read as the concatenation of two $N$-bit integers $x$, $y$, and after at most $cN$ many steps their product $z=xy$ is output in binary representation.

It is a rather detailed paper which would be silly to write up all here. See below.

Schönhage, A. (1980). Storage Modification Machines. SIAM Journal on Computing, 9(3), 490-508. https://doi.org/10.1137/0209036


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