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Solving efficiently a quadratic equation in a large finite field of characteristic two

I'm trying to solve efficiently a quadratic equation in the finite field $\text{GF}(2^{128})$ represented as $(\mathbb{Z}/2\mathbb{Z})[x] / (x^{128} + x^7 + x^2 + x + 1)$. Until now, I came across thi …
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