Polynomial multiplication over $\mathbb{F}_2$

The input is two polynomials over $\mathbb{F}_2$, each given as its $n$ coefficient bits. The output is the $2n-1$ coefficient bits of their product in $\mathbb{F}_2[x]$. Coefficients add without carries, so this is a different problem from integer multiplication. The machine is a deterministic Turing machine with a fixed number of tapes and a fixed alphabet. Its running time counts every step, including reading the input and writing the output. The quantity is $\kappa$ in a worst-case running time of $O(n\log^{1-\kappa}n)$. A higher $\kappa$ is better. The classic baseline is above $O(n\log n)$, so it has no $\kappa$. In it, $\log^*n$ is the number of times we take the logarithm to bring $n$ down to at most 1. Harvey and van der Hoeven (2022) reach $O(n\log n)$ only if an unproved hypothesis on primes in arithmetic progressions holds.

Classic baseline: $O(n\log n\,4^{\log^*n})$, published by David Harvey and Joris van der Hoeven in 2019.

These results are Claimed: their authors published them, and we have not checked them.

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