Niho type cross-correlation functions via dickson polynomials and Kloosterman sums

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2^k-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.