Cryptographic Hardness of Random Local Functions-Survey

Constant parallel-time cryptography allows performing complex cryptographic tasks at an ultimate level of parallelism, namely, by local functions that each of their output bits depend on a constant number of input bits. The feasibility of such highly efficient cryptographic constructions was widely studied in the last decade via two main research threads. The first is an encoding-based approach, developed in [1, 2], in which standard cryptographic computations are transformed into local computations via the use of special encoding schemes called randomized encoding of functions. The second approach, initiated by Goldreich [3], is more direct and it conjectures that almost all non-trivial local functions have some cryptographic properties. In this survey we focus on the latter approach. We consider random local functions in which each output bit is computed by applying some fixed d-local predicate P to a randomly chosen d-size subset of the input bits. Formally, this can be viewed as selecting a random member from a collection FP,n,m of dlocal functions where each member fG,P : {0, 1} → {0, 1} is specified by a d-uniform hypergraph G with n nodes and m hyperedges, and the i-th output of fG,P is computed by applying the predicate P to the d inputs that are indexed by the i-th hyperedge. In this talk, we will investigate the cryptographic hardness of random local functions. In particular, we will survey known attacks and hardness results, discuss different flavors of hardness (one-wayness, pseudorandomness, collision resistance, public-key encryption), and mention applications to other problems in cryptography and computational complexity. We also present some open questions with the hope to develop a systematic study of the cryptographic hardness of local functions.