A Lower Bound for the Nonlinearity of Exponential Welch Costas Functions

We study the nonlinearity of Exponential Welch Costas functions using the Fourier transform on Zn. Exponential Welch Costas functions are bijections from Zp−1 to Zp−1 defined using the exponential function of Zp, where p is an odd prime. Their linearity properties were recently studied by Drakakis, Requena, and McGuire, who conjectured that the absolute values of the Fourier coefficients of an Exponential Welch Costas function are bounded from above by O(p ), where is a small constant. In this paper, we establish an upper bound of order O( √ p log p), which is asymptotically strictly less than the bound conjectured by Drakakis et al.