Three Families of Monomial Functions With Three-Valued Walsh Spectrum

Let F-p be a finite field with p elements, where p is a prime. Let N >= 2 be an integer and d be the least positive integer satisfying p(d) equivalent to -1 (mod N). Let q = p(2sd) for some integers s. In some special cases, we obtain the explicit evaluation of the following exponential sums: S(a, b) = Sigma(x is an element of Fq*)zeta(Trq/p(ax((q-1)/N)+bx))(p). As applications, Walsh spectrums of the monomial functions Tr-q/p(x(((q-1)/N)))in three cases are investigated. Our results show that Walsh spectrums of the monomial functions have at most four, five, or seven distinct values. Furthermore, three families of the monomial functions with three-valued Walsh spectrums are presented, see Corollaries 12, 21, 31, and 32. Consequently, certain previously known results by Li and Yue and Moisio are extended.