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 n-variable quadratic Boolean functions, f"n","s(x)@[email protected]?"i"="1^nx"ix"i"+"s"-"1 for [email protected][email protected][email protected][email protected]?, which are homogeneous rotation symmetric, may not be affinely equivalent for fixed n and different choices of s. We show that their weights and nonlinearity are exactly characterized by the cyclic subgroup of Z"n. If ngcd(n,s-1), the order of s-1, is even, the weight and nonlinearity are the same and given by 2^n^-^1-2^n^2^+^g^c^d^(^n^,^s^-^1^)^-^1. If the order is odd, it is balanced and nonlinearity is given by 2^n^-^1-2^n^+^g^c^d^(^n^,^s^-^1^)^2^-^1.