When multiplying two numbers with each other, which are $n$-bit numbers, there are several algorithms like the one of Karatsuba ($O(n^{\log_2 3})$) and a new one doing it even better (Harvey - Van der Hoeven) with a complexity of $O(n \cdot \log n)$, which is much better, of course.

My question is now, if we a given a number $x$ and $m$ numbers $y_1, \ldots, y_m$, is there a faster algorithm than $O(m \cdot f(n))$, where $f(n)$ is the complexity of the fastest multiplication algorithm for just two numbers?

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