Projective Arithmetic Functional Encryption and Indistinguishability Obfuscation from Degree-5 Multilinear Maps

In this work, we propose a variant of functional encryption called projective arithmetic functional encryption (PAFE). Roughly speaking, our notion is like functional encryption for arithmetic circuits, but where secret keys only yield partially decrypted values. These partially decrypted values can be linearly combined with known coefficients and the result can be tested to see if it is a small value. We give a degree-preserving construction of PAFE from multilinear maps. That is, we show how to achieve PAFE for arithmetic circuits of degree d using only degree-d multilinear maps. Our construction is based on an assumption over such multilinear maps, that we justify in a generic model. We then turn to applying our notion of PAFE to one of the most pressing open problems in the foundations of cryptography: building secure indistinguishability obfuscation (iO) from simpler building blocks. iO from degree-5 multilinear maps. Recently, the works of Lin [Eurocrypt 2016] and LinVaikuntanathan [FOCS 2016] showed how to build iO from constant-degree multilinear maps. However, no explicit constant was given in these works, and an analysis of these published works shows that the degree requirement would be in excess of 30. The ultimate “dream” goal of this line of work would be to reduce the degree requirement all the way to 2, allowing for the use of well-studied bilinear maps, or barring that, to a low constant that may be supportable by alternative secure low-degree multilinear map candidates. We make substantial progress toward this goal by showing how to leverage PAFE for degree-5 arithmetic circuits to achieve iO, thus yielding the first iO construction from degree-5 multilinear maps. ∗Affiliated with Center for Encrypted Functionalities and Department of Computer Science, UCLA. This work is partially supported by grant #360584 from the Simons Foundation and the grants under Amit Sahai. †Affiliated with Center for Encrypted Functionalities and Department of Computer Science, UCLA.Research supported in part from a DARPA/ARL SAFEWARE award, NSF Frontier Award 1413955, NSF grants 1619348, 1228984, 1136174, and 1065276, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through the ARL under Contract W911NF-15-C-0205. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government.