Gold and Kasami-Welch functions, quadratic forms, and bent functions
Let \(n\) be odd, \(K={\mathbb F}_{2^n}\), \(Q(x)=\text{Tr}(x^{2^k+1}+x^{2^j+1})\) where \(0\leq j<k\). The authors prove that if \(\gcd(k+j,n)=\gcd(k-j,n)=1\), then the number of zeros of \(Q\) in \(K\) is \[ 2^{n-1}+\left(\frac2n\right)2^{\frac{n-1}{2}}, \] where \(\left(\frac2n\right)\) is the Jacobi symbol. This is used to evaluate the sign of some Walsh transforms in terms of a Jacobi symbol. These results are applied to the Walsh transforms of the Gold and Kasami-Welch functions. Gold functions yield bent functions when restricted to certain hyperplanes.
scientific article; zbMATH DE number 5864033