Square root for Galois fields $GF(2^m)$ Can we define a function similar to square root for $G = GF(2^m)$ (Galois field with $2^m$ elements) as $\sqrt{x} = y$ if $y^2 = y \cdot y = x$ ? For which elements $x \in G : \exists y \in G : y^2 = x$ this function would be defined?
Can I approach this question like this:
If we can generate all elements of $G$ except $0$ from another element $a$ as $a^k : k = 1 \ldots (2^m-1)$, then any $x \neq 0$ can be expressed as $x=a^r$ for some $r$ and $y$ would be also $y = a^s$ for some $s$. $y^2 = x$ means that $$r = 2s \mod 2^m-1$$ $$r,s \in 0 \ldots 2^m-2$$
It looks like I can find $s$ for any $r$ to satisfy equation. That would mean that there is a "square root" for any element in $G$, right?
PS: I'm looking into options to analyze streams of data (bytes, 16 bit or 32 bit integers) as part of another computational task, therefore only specific Galois fields are interesting for me: $GF(2^m)$. Be prepared that I can be way off in the field theory, that was very long time since I touched it - but any comments are welcome!

If GF(2^m) is small enough to use log and exponentiate tables, then square root of x can be implemented using tables, log[] and exp[]
L = log(x)
if (L is odd) L = L + 2^m - 1
E = L / 2
square root = exp(E)

If GF(2^m) is too large to use log and exponentiation tables, there is an alternative method. GF(2^m) is isomorphic to its field of square roots, since if a + b = c and a • b = d, then (a + b)^2 = a^2 + b^2 = c^2, and (a • b)^2 = a^2 • b^2 = d^2. The elements of GF(2^m) can be mapped to their square roots using an m by m matrix with 1 bit elements, where the values in the columns represent powers of the square root of 2 which is 2^((2^m)/2), which can be quickly calculated using exponentiation by squaring:
https://en.wikipedia.org/wiki/Exponentiation_by_squaring
Let σ = sqrt(2), then the square roots of 0 and powers of 2 are:
S[0] = 0
S[1] = 1
S[2] = σ
S[4] = S[2] • σ
S[8] = S[4] • σ
...

For the mapping matrix, the columns are indexed by powers of 2, and the values in the columns are the square roots of those powers of 2. For example, GF(2^8) with reducing polynomial x^8 + x^4 + x^3 + x^2 + 1 (hex 11D), treat an element of GF(2^8) as a matrix E[8][1], and the mapping matrix below as M[8][8], then the square root S[8][1] = M[8][8] • E[8][1] (carryless multiply since these are 1 bit elements of GF(2)).
80 40 20 10 08 04 02 01    value of E

 0  0  0  0  0  0  1  0
 1  0  0  0  0  0  0  0
 0  0  1  0  0  0  0  0
 1  0  0  0  1  0  0  0    mapping
 1  1  1  0  0  0  0  0    matrix
 1  0  1  1  1  0  1  0
 0  0  1  0  1  1  0  0
 0  0  0  0  1  0  1  1

5c 08 2e 04 17 02 85 01    value of S

