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

Abstract

Let r≡1(mod4) be a prime, m a positive integer, ϕ(rm)2 the multiplicative order of 2 modulo rm, and q=2ϕ(rm)2, where ϕ(⋅) is the Euler's function. Let γ be a primitive element of the finite field Fq and α=γq−1rm. In this paper, we shall explicitly calculate the exponential sums S(a)=∑i=0rm−1χ(aαi), a∈Fq, where χ is the canonical additive character of Fq. As applications, using elliptic curves and cyclotomic numbers over Fr, the three-valued Walsh spectrums of four classes of monomial functions fa(x)=Trq/2(axq−1rm) are determined.