On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2 (Q1003694)

In this paper the \(n\)-variable quadratic Boolean functions \(f_{n,s}=\sum _{i=1}^{n}x_{i}x_{i+s-1}\) for \(2\leq s\leq \lceil n/2\rceil \) are considered. They are homogeneous rotation symmetric, but may not be affinely equivalent for fixed \(n\) and different choices of \(s\). If \(k\) denotes gcd(\(n,s-1\)) it is shown that the weight and nonlinearity are the same and given by \(2^{n-1}-2^{n/2+k-1}\) if \(n/k\) is even and they are balanced (i. e., the weight is \(2^{n-1}\)) and nonlinearity is \(2^{n-1}-2^{(n+k)/2-1}\) otherwise.