In this paper, we propose a new notion of public key encryption scheme with authorized equality test (PKE-AET), which allows authorized users those who have warrants to test the equivalence between two messages, where the messages are encrypted using different public keys. Comparing with the existing researches, our PKE-AET provides two kinds of warrants that are referred to as receiver's warrants and cipher-warrants. The proposed PKE-AET is able to deal with the following complicated scenario: Assume that a receiver authorizes a receiver's warrant to a tester, which makes the tester be able to perform equality test on all of receivers’ ciphertext; on the other hand, if receiver authorizes a cipher-warrant corresponding to a specific ciphertext to the tester, then the tester only acquires the equality test on that particular ciphertext. The equality between two ciphertexts can be verified by the tester without decryption after he or she receives two warrants and varies their validations. Moreover, for security analysis, we define two types of adversaries and security notions for PKE-AET in the multi-user setting. Furthermore, we prove that our PKE-AET is one-way CCA secure against type-I adversaries and IND-CCA secure against type-II adversaries. Finally, the proposed scheme leads better efficiency than most of previous equality test schemes.

Traditional voting and bidding systems largely rely on paperwork and human resources throughout the voting process, which can incur high costs in terms of both time and money. Electronic voting and electronic bidding systems can be used to reduce costs, and many new systems have been introduced. However, most systems require a powerful and trusted third party to guarantee system integrity and security. With developments in blockchain technology, research has begun to highlight the core concept of decentralization. In this study, we introduce the first decentralized electronic voting and bidding systems based on a blockchain and smart contract. We also use cryptographic techniques such as oblivious transfer and homomorphic encryptions to improve privacy protection. Our proposed systems allow voters and bidders to participate in the opening phase and improve participant anonymity, the privacy of data transmission, and data reliability and verifiability. Moreover, compared with other electronic voting and bidding systems, our systems are safer and more efficient.