Verifiable Rotation of Homomorphic Encryptions

Abstract. Similar to verifiable shuffling (or, mixing), we consider the problem of verifiable rotating (and random re-encrypting) a given list of homomorphic encryptions. The offset by which the list is rotated (cyclic shift) should remain hidden. Basically, we will present zero-knowledge proofs of knowledge for the existence of a rotation offset and re-encryption exponents, which define how the input list is transformed into the output list. We also briefly address various applications of verifiable rotators, ranging from ‘fragile mixing ’ as introduced by Reiter and Wang at CCS’04 to applications in protocols for secure multiparty computation and voting. We present two new, efficient protocols. Our first protocol is quite elegant and involves the use of the Discrete Fourier Transform (and also the FFT algorithm), and works under some reasonable conditions. We believe that this is the first time that Fourier Transforms are used to construct an efficient zero-knowledge proof of knowledge. Our second protocol is more general (requiring no further conditions) and only slightly less efficient than the DFT-based protocol. Unlike the previously best protocol by Reiter and Wang, however, which relies on extensive use of verifiable shuffling as a building block (invoking it four times as a sub-protocol), our construction is direct and its performance is about equal to the performance of a single run of the best protocol for verifiable shuffling.