Oblivious Linear Group Actions and Applications

Abstract

In this paper we propose efficient two-party protocols for obliviously applying a (possibly random) linear group action to a data set. Our protocols capture various applications such as oblivious shuffles, circular shifts, matrix multiplications, to name just a few. A notable feature enjoyed by our protocols, is that they admit a round-optimal (more precisely, one-round) online computation phase, once an input-independent off-line computation phase has been completed. Our oblivious shuffle is the first to achieve a round-optimal online phase. The most efficient instantiations of our protocols are obtained in the so-called client-aided client-server setting, where the offline phase is run by a semi-honest input party (client) who will then distribute the generated correlated randomness to the computing parties (servers). When comparing the total running time to the previous best two-party oblivious shuffle protocol by Chase et al. (Asiacrypt 2020), our shuffle protocol in this client-aided setting is up to 105 times and 152 times faster, in the LAN and WAN setting, respectively. We additionally show how the Chase et al. protocol (which is a standard two-party protocol) can be modified to leverage the advantages of the client-aided setting, but show that, even doing so, our scheme is still two times faster in the online phase and 1.34 times faster in total on average. An additional feature of our protocols is that they allow to re-invoke a previously generated group action, or its inverse, in subsequent runs. This allows us to utilize randomize-then-reveal techniques, which are crucial for constructing efficient protocols in complex applications. As an application, we construct a new oblivious sorting protocol implementing radix sort. Our protocol is based on a similar approach to the three-party protocol by Chida et al. (IACR ePrint 2019/965), but using our oblivious shuffle as a building block as well as various optimizations, we obtain a two-party protocol (in the client-aided setting) with improved online running time and a reduced number of rounds. As other applications, we also obtain efficient protocols for oblivious selection, oblivious unit-vectorization, oblivious multiplexer, oblivious polynomial evaluation, arithmetic-to-boolean share conversions, and more.