On the cross-correlation of Golomb Costas permutations

In the most interesting case of safe prime powers q, G\'omez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of \1,2,…,q-2\ of size φ(q-1) has maximal cross-correlation of order of magnitude at most q1/2. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemer\'edi-Trotter theorem for finite fields.