An upper bound for the linearity of Exponential Welch Costas functions

The maximum correlation between a function and affine functions is often called the linearity of the function. In this paper, we determine an upper bound for the linearity of Exponential Welch Costas functions using Fourier analysis on Zn. Exponential Welch Costas functions are bijections on Zp−1, where p is an odd prime, defined using an exponential function of Zp. Their linearity properties were recently studied by Drakakis, Requena, and McGuire (2010) [1] who conjectured that the linearity of an Exponential Welch Costas function on Zp−1 is bounded from above by O(p0.5+ϵ), where ϵ is a small constant. We prove that the linearity is upper bounded by , which is asymptotically strictly less than what was previously conjectured.