Secret sharing scheme is a key component of distributed cryptosystems. In its basic form, secret sharing schemes can tolerate honest but curious adversary. But, in modern open system environment, adversary can behave maliciously i.e., the adversary can do anything according to his available computational resources. To get rid of such adversary, cheating identifiable (multi) secret sharing scheme plays an important role. Informally, cheating identifiable (multi) secret sharing scheme can identify the cheating participants, who are under the control of malicious adversary, and recover the correct secret whenever possible. However, to achieve unconditional security against such adversary, share size should be at least equal to the size of the secret. As a result, the need for computational notion of security of such schemes, which can accommodate smaller share size, has been felt over the years, specially in case of multi-secret sharing schemes. In this paper, we propose a notion of security for computationally secure cheating identifiable multi-secret sharing scheme for general access structure along with a construction which is secure under this new notion. © Springer International Publishing Switzerland 2015.