Cross-correlation of M-sequences, exponential sums and dickson polynomials

Let s(t) and s(dt) be two binary m-sequences of the same period 2^m − 1 where gcd(d,2^m − 1)=1 . The cross-correlation between these two m-sequences is defined to be where the summation is over a full period t = 0,1,…2^m−2. The main problem is to find the distribution of the cross-correlation i.e., the values that occur and their multiplicity when τ ranges through 0,1…,2^m − 2. The cross-correlation between m-sequences is a well studied problem during the last 40 years. A survey over known results as well as an overview of some interesting remaining open problems is presented. In particular new recent results are presented for the cross-correlation of sequences with decimations on the form d = (2^k + 1)/2^r + 1) using connections to Dickson polynomials and calculations of some special exponential sums.