Is there a way to map a characteristic-2 field $GF(2^m)$ to a larger prime field $GF(p)$, perform arithmetic in that field, and then map it back to the original field as though I had stayed in the original field?
Background: FPGA chips have a lot of fast multipliers built into the silicon, but when doing multiplications in, say, $GF(2^{10})$, these are useless to me and go unused. I could use these for multiplications in a non-extension prime field p, using one multiplier for the multiplication and another for reduction mod p. I'd like to map the original values from one field to another, do matrix multiplications, and then map that answer back to the extension field.
If you map some nonzero element $x \in GF(2^m)$ to $y \in GF(p)$, you'll have $y + y \ne 0$ but $x + x = 0$, so the result of arithmetic on the images does not map back to the result of arithmetic in the original field.
Similarly for multiplication, $x^{2^m} = x$ but $y^{2^m} \ne y$.