Simple PCPs with poly-log rate and query complexity

Abstract

We give constructions of probabilistically checkable proofs (PCPs) of length n . poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters.Previous constructions of short PCPs (from [5]to [9]) relied extensively on properties of low degree multi-variate polynomials. In contrast, our constructions rely on new problems and techniques revolving around the properties of codes based on high degree polynomials in one variable (also known as Reed-Solomon codes). We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a Reed-Solomon codeword, i.e., a univariate polynomial of specified degree. This reduction is simpler than the corresponding steps in previous reductions, and gives a new alternative to using the popular sum-check protocol. We then give a new PCP for the special task of proving that a function is close to being a Reed-Solomon codeword. This step of the construction is by a self-contained recursion, and the only ingredient needed in the analysis is the bi-variate low-degree test of Polischuk and Spielman[27].Note that our constructions yield LTCs first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite (and less natural) direction of getting LTCs from PCPs.