Paper 2021/842
PCPs and Instance Compression from a Cryptographic Lens
Liron Bronfman and Ron D. Rothblum
Abstract
Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary's power in other information theoretic proofsystems and show how to use this assumption to bypass impossibility results. We first consider the question of constructing succinct PCPs. These are PCPs whose length is polynomial only in the length of the original NP witness (in contrast to standard PCPs whose length is proportional to the nondeterministic verification time). Unfortunately, succinct PCPs are known to be impossible to construct under standard complexity assumptions. Assuming the subexponential hardness of the learning with errors (LWE) problem, we construct succinct probabilistically checkable arguments or PCAs (Zimand 2001, Kalai and Raz 2009), which are PCPs in which soundness is guaranteed against efficiently generated false proofs. Our PCA construction is for every NP relation that can be verified by a smalldepth circuit (e.g., SAT, clique, TSP, etc.) and in contrast to prior work is publicly verifiable and has constant query complexity. Curiously, we also show, as a proofofconcept, that such publiclyverifiable PCAs can be used to derive hardness of approximation results. Second, we consider the notion of Instance Compression (Harnik and Naor, 2006). An instance compression scheme lets one compress, for example, a CNF formula $\varphi$ on $m$ variables and $n \gg m$ clauses to a new formula $\varphi'$ with only $poly(m)$ clauses, so that $\varphi$ is satisfiable if and only if $\varphi'$ is satisfiable. Instance compression has been shown to be closely related to succinct PCPs and is similarly highly unlikely to exist. We introduce a computational analog of instance compression in which we require that if $\varphi$ is unsatisfiable then $\varphi'$ is effectively unsatisfiable, in the sense that it is computationally infeasible to find a satisfying assignment for $\varphi'$ (although such an assignment may exist). Assuming the same subexponential LWE assumption, we construct such computational instance compression schemes for every boundeddepth NP relation. As an application, this lets one compress $k$ formulas $\phi_1,\dots,\phi_k$ into a single short formula $\phi$ that is effectively satisfiable if and only if at least one of the original formulas was satisfiable.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint. MINOR revision.
 Keywords
 PCPsuccinct argumentsinstance compression
 Contact author(s)

rothblum @ cs technion ac il
br @ cs technion ac il  History
 20210621: received
 Short URL
 https://ia.cr/2021/842
 License

CC BY
BibTeX
@misc{cryptoeprint:2021/842, author = {Liron Bronfman and Ron D. Rothblum}, title = {{PCPs} and Instance Compression from a Cryptographic Lens}, howpublished = {Cryptology {ePrint} Archive, Paper 2021/842}, year = {2021}, url = {https://eprint.iacr.org/2021/842} }