How to find $z$ when deserializing elliptic curve?

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In §2.3.4 of Standards for Efficient Cryptography 1 (SEC 1), the authors define the following step in deserializing elliptic-curve points that were serialized in the format given in §2.3.3 (emphasis added):

If $q = 2^m$ and $x_P\neq 0$, compute the field element $\beta=x_P+a+b x_P^{-2}$ in $\mathbb{F}_{2^m}$, and find an element $z=z_{m-1}x^{m-1}+\cdots+z_1 x+z_0$ such that $z^2+z=\beta$ in $\mathbb{F}_{2^m}$. Output “invalid” and stop if no such $z$ exists, otherwise set $y_P=x_Pz$ in $\mathbb{F}_{2^m}$ if $z_0=\tilde{y}_P$, and set $y_P=x_P{({z+1})}$ in $\mathbb{F}_{2^m}$ if $z_0\neq\tilde{y}_P$.

What method can be used to compute $z$ in this context?

kelalaka avatar
in flag
I think this is a dupe [Solving Quadratic equations in Galois Field (2^163)](, of course if you need to find, use SageMath instead of solving by hand.
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