The cross-correlation between two maximum-length sequences (m-sequences) of the same period has been studied since the 1960s. A conjecture by Helleseth states that the cross-correlation between any two m-sequences takes on the value -1 for at least one shift provided that the decimation between the two m-sequences obeys d=1 (mod p-1) where the alphabet is GF(p).

One important problem in finite fields is to find complete permutation polynomials. These are polynomials c(x) where both c(x) and c(x)+x are permutation polynomials.

Recently several new results on complete monomial permutation polynomials (i.e. of the form ax^d) have appeares. We survey some of these results and discuss connections between the -1 conjecture and complete monomial permutation polynomials.