Abstract: It is known that Boolean functions used in stream and block ciphers should have good cryptographic properties to resist the existing efficient attacks. The number of linearly independent low degree annihilators of a given Boolean function and of its complement function is an important parameter for evaluating the complexity of algebraic attacks on the systems using this Boolean function. The dimensions of vector spaces of annihilators for Boolean functions have received much attention in cryptographic literature. According to one-to-one correspondence between Maiorana-McFarland's (M-M) Bent functions and Boolean permutations, a family of Boolean functions are presented. Moreover, it is shown that the presented family of Boolean functions is linearly independent. In addition, it is known that every nonzero linear combination of a Boolean permutation is a balanced Boolean function. On the basis of the above facts, a new upper bound on the dimension of vector spaces of annihilators with prescribed degrees of a special M-M Bent function and of its complement is proposed. As far as the special M-M Bent functions are concerned, the new upper bound is less than the known ones. Furthermore, the new upper bound for all M-M Bent functions can be obtained.