Summary: The cryptographic properties of the balanced \(H\)-Boolean functions are deeply studied and some theorems of multivariate-balanced \(H\)-Boolean functions are obtained. It is shown that three-variate-balanced \(H\)-Boolean functions are absolutely not of correlation-immunity while for four-variate-balanced \(H\)-Boolean functions, their first-order correlation-immunity can be obtained by means of construction, which will further reveal the fine cryptographic properties of balanced \(H\)-Boolean functions and lay a foundation for studying the cryptographic properties of the Boolean functions, guaranteeing the safety and anti-attack ability of the cryptosystem.