On the concrete efficiency of probabilistically-checkable proofs

Probabilistically-Checkable Proofs (PCPs) form the algorithmic core that enables fast verification of long computations in many cryptographic constructions. Yet, despite the wonderful asymptotic savings they bring, PCPs are also the infamous computational bottleneck preventing these powerful cryptographic constructions from being used in practice. To address this problem, we present several results about the computational efficiency of PCPs. We construct the first PCP where the prover and verifier time complexities are quasi-optimal (i.e., optimal up to poly-logarithmic factors). The prover and verifier are also higly-parallelizable, and these computational guarantees hold even when proving and verifying the correctness of random-access machine computations. Our construction is explicit and has the requisite properties for being used in the cryptographic applications mentioned above.

Next, to better understand the efficiency of our PCP, we propose a new efficiency measure for PCPs (and their major components, locally-testable codes and PCPs of proximity). We define a concrete-efficiency threshold that indicates the smallest problem size beyond which the PCP becomes "useful", in the sense that using it is cheaper than performing naive verification (i.e., rerunning the computation); our definition accounts for both the prover and verifier complexity.

We then show that our PCP has a finite concrete-efficiency threshold. That such a PCP exists does not follow from existing works on PCPs with polylogarithmic-time verifiers.

As in [Ben-Sasson and Sudan, STOC '05], PCPs of proximity for Reed-Solomon (RS) codes are the main component of our PCP. We construct a PCP of proximity that reduces the concrete-efficiency threshold for testing proximity to RS codes from 2⁶⁸³ in their work to 2⁴³, which is tantalizingly close to practicality.