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### Square root of polynomial modulo another irreducible polynomial

Hello,

If I'm not wrong, it is always possible to compute the square root of a polynomial $P$ modulo an irreducible polynomial $g$ when the base field is in $GF(2^m)$, i.e. find $Q \in GF(2^m)$ such that $Q^2 \equiv P \mod g$. Indeed, the operation $Q \rightarrow Q^2 \pmod g$ should be linear (because we are in $GF(2^m)$) so an idea would be to compute the matrix $T$ that perform this operation, and then invert it. I tried the sagemath $P.sqrt()$ method, but the problem is that because it does not take into account the modulo, it fails most of the time when the polynomial has some terms with odd power of $X$.

Any idea?

Thanks!

### Square root of polynomial modulo another irreducible polynomial

Hello,

If I'm not wrong, it is always possible to compute the square root of a polynomial $P$ modulo an irreducible polynomial $g$ when the base field is in $GF(2^m)$, i.e. find $Q \in GF(2^m)$ such that $Q^2 \equiv P \mod g$. Indeed, the operation $Q \rightarrow Q^2 \pmod g$ should be linear (because we are in $GF(2^m)$) so an idea would be to compute the matrix $T$ that perform this operation, and then invert it. it, but I'd like to find an embedded operation in sage. I tried the sagemath $P.sqrt()$ method, but the problem is that because it does not take into account the modulo, it fails most of the time when the polynomial has some terms with odd power of $X$.

Any idea?

Thanks!