What are known results regarding the distribution of the powers of a primitive element (generator of the multiplicative group) of a finite field? Specifically, compare the ordered list of ascending powers of the generator with the natural ordering of the field.

If it helps, this is for some research I am doing in cryptography, involving analysis of an algorithm that is dependent on the powers of a multiplicative generator of a finite field of order $2^n$. So I would be happy to see such theorems on just binary finite fields.

specificirreducible polynomial and take the quotient of the relevant polynomial ring, the natural order becomes clear. I should have said that for prime powers, we must fix what we are modding out by. – Seraj Feb 2 '13 at 19:47