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?
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$\begingroup$ Why do you think the page is talking about bit complexity? $\endgroup$– Yuval FilmusSep 9, 2017 at 10:21
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$\begingroup$ Because when speaking of words it takes $O(1)$ time. $\endgroup$– rus9384Sep 9, 2017 at 11:05
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1 Answer
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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
