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Understanding Intel's algorithm for reducing a polynomial modulo an irreducible polynomial

For completeness, I'll flesh out Yuval's answer a bit more through an example of multiplication of two polynomials $A$ and $B$ in $\text{GF}(2^{16})$. … .$$ The multiplication of $A$ and $B$ over $\text{GF}(2)$ is then $$C = [0000|0000|0000|0111 | 0101|0010|0000|1010].$$ We then compute for $\text{GF}(2^{16})$ the polynomial $$q^+ = x^{32}+x^{21}+x^{19 …
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