Functional Encryption (FE) allows users to extract specific function-related information from encrypted data while preserving the privacy of the underlying plaintext. Though significant research has been devoted to developing secure and efficient Multi-Input Functional Encryption schemes supporting diverse functions, there remains a noticeable research gap in the development of verifiable FE schemes. Functionality and performance have received considerable attention, however, the crucial aspect of verifiability in FE has been relatively understudied. Another important aspect that prior research in FE with outsourced decryption has not adequately addressed is the fairness of the data-for-money exchange between a curator and an analyst. This paper focuses on addressing these gaps by proposing a verifiable FE scheme for inner product computation. The scheme not only supports the multi-client setting but also extends its functionality to accommodate multiple users -- an essential feature in modern privacy-respecting services. Additionally, it demonstrates how this FE scheme can be effectively utilized to ensure fairness and atomicity in a payment protocol, further enhancing the trustworthiness of data exchanges.

As data sharing has become one of the most popular services offered by cloud storage, designing public auditing mechanisms for integrity of shared data becomes more important. Two problems which arise in shared data auditing include preserving users identity and collusion resistant revocation of users. When data stored at the cloud is shared among a group of users, different users may modify and sign different data blocks which leaks signer identities to the public verifier. Also, when a user is revoked from the group, signatures generated by this user should be re-signed by the cloud server using re-signature keys. In addition, collusion of cloud server and the revoked user should leak no information about the private key of other users. In this paper, by employing a new proxy re-signature scheme, we propose a public shared data auditing mechanism that provides identity privacy and collusion resistant user revocation, simultaneously. The proposed protocol requires only lightweight computations at the user side for signing data blocks in real-time online phase. Moreover, our protocol supports large dynamic group of users, batch auditing and dynamic data operations. Experimental results demonstrate excellent efficiency of our scheme in comparison to the state of the art.