Three Families of Monomial Functions With Three-Valued Walsh Spectrum
Yansheng Wu, Qin Yue, Fengwei Li · IEEE Transactions on Information Theory · 2018
Let$\Bbb F_{p}$be a finite field with$p$elements, where$p$is a prime. Let$N \ge 2$be an integer and$d$be the least positive integer satisfying$p^{d} \equiv -1 \pmod N$. Let$q = p^{2sd}$for some integers$s$. In some special cases, we obtain the explicit evaluation of the following exponential sums:$S(a,b)=\sum _{x\in \Bbb F_{q}^{*}}\zeta _{p}^{ \mathrm {Tr}_{q/p}(ax^{(({q-1})/{N})}+bx)}$. As applications, Walsh spectrums of the monomial functions$\mathrm {Tr}_{q/p}(x^{(({q-1})/{N})})$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, seeCorollaries 12,21,31, and32. Consequently, certain previously known results by Li and Yue and Moisio are extended.