An Attribute Based Searchable Encryption Scheme (ABSE) is a public key encryption with keyword search (PEKS) where each user owns a set of attributes, and the senders decide on a policy. The policy is a function of these attributes expressed as a predicate and determines, among the users of the system, who is eligible to decrypt and search the ciphertext. Only members who own sufficient attributes to satisfy that policy can send the server a valid search query. In our work we introduce the concept of a secure ABSE by defining the functionalities and the relevant security notions.