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.