Graded Encoding, Variations on a Scheme

Update (Oct 2015): The key-agreement protocols that are described (or alluded to) in sections 6,7 are broken. Thanks to Yupu Hu and Jean-Sébastien Coron for pointing out to me various attacks, these attacks (and their extensions) will be described in separate reports in the coming weeks/months. In this note we provide a more-or-less unified framework to talk about the functionality and security of graded encoding schemes, describe some variations of recent schemes, and discuss their security. In particular we describe schemes that combine elements from both the GGH13 scheme of Garg, Gentry and Halevi (EUROCRYPT 2013) and the GGH15 scheme of Gentry, Gorbunov and Halevi (TCC 2015). On one hand, we show how to use techniques from GGH13 in the GGH15 construction to enable encoding of arbitrary plaintext elements (as opposed to only small ones) and to introduce “levels/subsets ” (e.g., as needed to implement straddling sets). On the other hand, we show how to modify the GGH13 scheme to support graph-induced constraints (either instead of, or in addition to, the levels from GGH13). Turning to security, we describe zeroizing attacks on the GGH15 scheme, similar to thos