  Nonlinearity of rotation symmetric Boolean functions is an important topic on
cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we
investigate the following question: Is the nonlinearity of the quartic rotation
symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal
to its weight? We introduce some new simple sub-functions and develop new
technique to get several recursive formulas. Then we use these recursive
formulas to show that the nonlinearity of the quartic rotation symmetric
Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ is the same as
its weight. So we answer the above question affirmatively. Finally, we
conjecture that if $l\ge 4$ is an integer, then the nonlinearity of the
rotation symmetric Boolean function generated by the monomial
$x_0x_ex_{2e}...x_{le}$ equals its weight.
