Verifiable Outsourcing Algorithms for Modular Exponentiations with Improved Checkability

The problem of securely outsourcing computation has received widespread attention due to the development of cloud computing and mobile devices. In this paper, we first propose a secure verifiable outsourcing algorithm of single modular exponentiation based on the one-malicious model of two untrusted servers. The outsourcer could detect any failure with probability 1 if one of the servers misbehaves. We also present the other verifiable outsourcing algorithm for multiple modular exponentiations based on the same model. Compared with the state-of-the-art algorithms, the proposed algorithms improve both checkability and efficiency for the outsourcer. Finally, we utilize the proposed algorithms as two subroutines to achieve outsource-secure polynomial evaluation and ciphertext-policy attributed-based encryption (CP-ABE) scheme with verifiable outsourced encryption and decryption.