On the Black-box Use of Somewhat Homomorphic Encryption in NonInteractive Two-Party Protocols

Abstract

In this work, we develop a methodology for determining the communication required to implement various two-party functionalities noninteractively. In the particular setting on which we focus, the protocols are based upon somewhat homomorphic encryption, and furthermore, they treat the homomorphic properties as a black box. In this setting, we develop lower bounds which give a smooth trade-off between the communication complexity and the “expressiveness” of the cryptosystem---the latter being measured in terms of the depth of the arithmetic circuits that can be evaluated on ciphertext. Given the current state of the art in homomorphic encryption, this trade-off may also be viewed as one between communication and computation, since at present, more expressive cryptosystems are markedly less efficient. We then apply this methodology to place lower bounds on a number of cryptographic protocols including private information retrieval writing and private keyword search. Our work provides a useful “litmus test” ...