I need to solve the usual cubic equation $x^3 + ax^2 + bx + c = 0$ over a finite field $GF(2^n)$. This is to avoid doing a brute-force Chien search in a BCH decoder.

I read in a paper about an easy way to reduce the equation to $z^3 + z + k = 0$ where $k = (a b + c) / (a^2 + b)^{3/2}.$ Then a lookup table can be used for all the values of $k$. However as far as I can tell this approach will not work when $(a^2 + b)$ happens to not have a square root in the field. An example for $GF(2^5)$ would be $x^3 + 08*x + 1a = 0.$

Can someone help?

Regards,

-Dimitri