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

We improve parts of the results of [T. W. Cusick, P. Stanica, Fast evaluation, weights and nonlinearity of rotation-symmetric functions, Discrete Mathematics 258 (2002) 289–301; J. Pieprzyk, C. X. Qu, Fast hashing and rotation-symmetric functions, Journal of Universal Computer Science 5 (1) (1999) 20–31]. It is observed that the nn-variable quadratic Boolean functions, fn,s(x)≔∑i=1nxixi+s−1 for 2≤s≤⌈n2⌉, which are homogeneous rotation symmetric, may not be affinely equivalent for fixed nn and different choices of ss. We show that their weights and nonlinearity are exactly characterized by the cyclic subgroup 〈s−1〉〈s−1〉 of ZnZn. If ngcd(n,s−1), the order of s−1s−1, is even, the weight and nonlinearity are the same and given by 2n−1−2n2+gcd(n,s−1)−1. If the order is odd, it is balanced and nonlinearity is given by 2n−1−2n+gcd(n,s−1)2−1.