New Proofs of (New) Direct Product Theorems

Direct product theorems are formal statements of the intuition, “It is harder to solve multiple instances of a problem than to solve a single instance.” For this talk, the amplification of hardness will be in reducing the probability of success for a feasible algorithm. So a direct product result for circuits might say, “If no small circuit can compute Boolean function f(x) with probability at least 1 − δ for a random x, then no circuit of a slightly smaller size can compute f(x1)...f(xk) with even a small probability of success.” Besides average-case complexity, direct product results have been very important in derandomization (making a somewhat hard problem into a “pseudo-random” one) and in cryptography, where the goal is to create problems that are reliably hard for the attacker to solve. They have also had non-obvious applications in other areas of complexity. While intuitive, such results are frequently non-trivial. One reason for this is that there are other circumstances where the intuition is incorrect, and many instances are not proportionally harder. For example, the above theorem fails when “small circuit” is replaced by “fast probabilistic algorithm”. Another example of a setting where direct products fail are parallel repetition for multiple round protocols and for non-verifiable puzzles (see: BIN97, CHS05, PW07). Proofs of direct product theorems, even for circuit complexity, are interesting in and of themselves. There seem to be many distinct proofs, each of which generalizes in different ways or can be emulated in other settings. Following Trevisan, different proof techniques for direct product can be viewed as different approximate list-decoding algorithms. In this talk, we will show how different versions of direct product theorem at least seem to require different proof techniques. 1. As mentioned above, the strong statement of the direct product theorem fails in uniform contexts such as probabilistic algorithms. However, following Trevisan’s view of direct products as codes, we can identify the reason for this: approximate unique decoding is not possible, only list-decoding. That makes the size of the list (equivalently, the “non-uniformity”) a parameter of direct product theorems that can be minimized. We show how to reduce this list size