A Hierarchical Key Assignment Scheme (HKAS) is a method to assign some private information and secret keys to a set of classes in a partially ordered hierarchy, so that the private information of a higher class together with some public information can be used to derive the keys of all classes lower down in the hierarchy. Historically, HKAS has been introduced to enforce multi-level access control, where it can be safely assumed that the public information is made available in some authenticated form. Subsequently, HKAS has found application in several other contexts where, instead, it would be convenient to certify the trustworthiness of public information. Such application contexts include key management for IoT and for emerging distributed data acquisition systems such as wireless sensor networks. In this paper, motivated by the need of accommodating this additional security requirement, we first introduce a new cryptographic primitive: Verifiable Hierarchical Key Assignment Scheme (VHKAS). A VHKAS is a key assignment scheme with a verification procedure that allows honest users to verify whether public information has been maliciously modified to induce an honest user to obtain an incorrect key. Then, we design and analyse VHKASs which are provably secure. Our solutions support key update for compromised secret keys by making a limited number of changes to public and private information.