On the cross-correlation of Golomb Costas permutations

In the most interesting case of safe prime powers $q$, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of $\{1,2,\dots ,q-2\}$ of size $\phi (q-1)$ has maximal cross-correlation of order of magnitude at most $q^{1\mathrm{/}2}$. 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 Szemerédi-Trotter theorem for finite fields.