Several classes of exponential sums and three-valued Walsh spectrums over finite fields

Let r 1(mod 4) be a prime, ma positive integer, phi(r(m))/2 the multiplicative order of 2 modulo r(m), and q = 2 phi(r(m))/2, where phi(center dot) is the Euler's function. Let gamma be a primitive element of the finite field F-q and alpha = gamma q-1/r(m). In this paper, we shall explicitly calculate the exponential sums S(a) = Sigma(rm-1)(i=0) chi(a alpha(i)), a is an element of F-q, where chi .is the canonical additive character of F-q. As applications, using elliptic curves and cyclotomic numbers over F-r, the three-valued Walsh spectrums of four classes of monomial functions f(a)(x) = Tr-q/2(ax(q-1/rm)) are determined. (c) 2022 Elsevier Inc. All rights reserved.