TCS+ Talk

Abstract:  A recent line of work shows how to construct indistinguishability obfuscation under two assumptions: (a) that there exist k-linear maps for some constant k; and (b) that certain random O(k)-constraint satisfaction problems (CSPs) are hard in an appropriate sense. The latest of these works (by Lin and Tessaro) assumes the existence of 3-linear maps and the hardness of certain random 3-CSPs. We have 1-linear maps since the 1970s; 2-linear maps since the 1990s; but the existence of 3-linear maps is wide open. On the other hand, we do have reasonable constructions of "secure" random 3-CSPs. The first part of the talk will describe these developments.

Much more surprising was a result (from the same work of Lin and Tessaro) which showed a construction from 2-linear maps and the hardness of random 2-CSPs over a large alphabet. Overnight, the burden of existence of IO went from the question of whether 3-linear maps exist to the completely unrelated question of whether random 2-CSPs over large alphabets is hard. In a nutshell, they require the existence of pseudo-random generators G: \Sigma^n \to {0,1}^m for some poly(n)-size alphabet \Sigma where each output bit depends on at most two input alphabet symbols, and which achieve sufficiently large stretch. In the second part of the talk, we will present a polynomial-time algorithm that breaks these random CSPs.

Based on joint work with Alex Lombardi (MIT) and Rachel Lin (UCSB).