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In this video, we delve into the fascinating world of big number multiplication and explore how computers perform this task ...
For such enormous numbers, even Karatsuba's algorithm is too slow. A real breakthrough came in 1971 with the work of the German mathematicians Arnold Schönhage and Volker Strassen.
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Karatsuba’s Algorithm: The Mathematical Breakthrough that ... - MSN

Known today as Karatsuba’s Algorithm, this breakthrough challenged the long-held belief that multiplying two n-digit numbers required n² steps.
[Nemean] has a video explaining a better multiplication algorithm known as Karatsuba’s algorithm and it is actually quite clever. You can see the video below.