Galois Field multiplication instead of Diffie Hellmans discrete logarithm I am wondering if the inversion of multiplication of polynomials is equally hard as the discrete logarithm problem used for key exchange. Or are there algorithms that weaken such an usage. I understand that it is somewhat easy to factorize if one omits the division by an irreducible polynomial.

I cannot find any comparison for the hardness of


multiplicative inverse in GF(2^n) mod (some irreducible polynomial)
Diffie Hellman using exponents of g^x mod p
elliptic curves


Only for the last two I was able to find some comparison which favors elliptic curves over the discrete logarithm problem as the key length is about 1/12 as opposed to Diffie Hellman for the same security.
The inverse is easy, it can be done by the extended Euclidean algorithm, thus polynomial in complexity compared to the others which are exponential.