In this article we apply some number theoretical techniques to derive results on Boolean functions. We apply Stickelberger’s theorem on 2-adic valuations of Gauss sums to the Kasami-Welch functions trL(x4k−2k+1) on F2n , where n is odd and (k, n)= 1. We obtain information on the Fourier spectrum, including a characterization of the support of the Fourier transform. One interesting feature is that the behaviour is different for different values of k. We also apply the Gross-Koblitz formula to the Gold functions trL(x2k+1).