Since the first candidate construction of multilinear map by Garg, Gentry and Halevi, multilinear maps have been used as building blocks of a lot of exciting cryptographic applications, including indistinguishability obfuscation. However current instantiations rely on heuristic security analyses. The zero testing procedure, at the heart of the current mutilinear maps candidates, has recently been used to mount powerful attacks. Numerous hardness assumptions and constructions from the literature were therefore broken.
The candidate multilinear maps over the integers have recently been revisited by Coron, Lepoint and Tibouchi (Crypto 2015): by modifying the zero-testing procedure, known attacks seem thwarted. In this talk, we will review the design of the initial CLT scheme and its vulnerabilities. Then we will present and discuss the Crypto 2015 modification of CLT, and especially what are its hardness assumptions and expectable security.