Indistinguishability Obfuscation Without Maps: Attacks and Fixes for Noisy Linear FE

摘要

Candidates of Indistinguishability Obfuscation (iO) can be categorized as "direct" or "bootstrapping based". Direct constructions rely on high degree multilinear maps and provide heuristic guarantees, while bootstrapping based constructions rely, in the best case, on bilinear maps as well as new variants of the Learning With Errors (LWE) assumption and pseudorandom generators. Recent times have seen exciting progress in the construction of indistinguishability obfuscation (iO) from bilinear maps (along with other assumptions). As a notable exception, a recent work by Agrawal provided a construction for iO without using any maps. This work identified a new primitive, called Noisy Linear Functional Encryption (NLinFE) that provably suffices for iO and gave a direct construction of NLinFE from new assumptions on lattices. While a preliminary cryptanalysis for the new assumptions was provided in the original work, the author admitted the necessity of performing significantly more cryptanalysis before faith could be placed in the security of the scheme. Moreover, the author did not suggest concrete parameters for the construction. In this work, we fill this gap by undertaking the task of thorough crypt-analytic study of NLinFE. We design two attacks that let the adversary completely break the security of the scheme. Our attacks are completely new and unrelated to attacks that were hitherto used to break other candidates of iO. To achieve this, we develop new cryptanalytic techniques which (we hope) will inform future designs of the primitive of NLinFE. From the knowledge gained by our cryptanalytic study, we suggest modifications to the scheme. We provide a new scheme which overcomes the vulnerabilities identified before. We also provide a thorough analysis of all the security aspects of this scheme and argue why plausible attacks do not work. We additionally provide concrete parameters with which the scheme may be instantiated. We believe the security of NLinFE stands on significantly firmer footing as a result of this work.