Interactive proofs of proximity (IPPs) offer ultra-fast approximate verification of assertions regarding their input, where ultra-fast means that only a small portion of the input is read and approximate verification is analogous to the notion of approximate decision that underlies property testing (PT). Specifically, in an IPP, the prover can make the verifier accept each input in the property, but cannot fool the verifier into accepting an input that is far from the property (except for with small probability).
The verifier in an IPP system engages in two very different types of activities: interacting with an untrusted prover, and querying its input. The classic definition of IPP allows for arbitrary coordination between these two activities, but keeping them separate is both conceptually interesting and necessary for important applications such as addressing temporal considerations (i.e., at what time is each of the services available) and facilitating the construction of zero-knowledge schemes. In this thesis we embark on a systematic study of IPPs where the queries should not be affected by the interaction with the prover, termed Proof-Oblivious (PO) queries. By assigning the query and interaction activities to separate modules, we can consider different limitations on their coordination, and present a hierarchy of PO queries IPPs.
The most strict limitation requires these activities to be totally isolated from one another; they just feed their views to a separate deciding module. We show that such systems can be efficiently emulated by standard testers.
In the other extreme, we only disallow information to flow from the interacting module to the querying module, but allow free information flow in the other direction. We show that extremely efficient one-round (i.e., two-message) systems of such type can be used to verify properties that are extremely hard to test (without the help of a prover). That is, the complexity of verifying can be polylogarithmic in the complexity of testing. This stands in contrast to MAPs (a $1/2$-round system) in which proof-oblivious queries are as limited as our isolated model.
Our main focus is on an intermediate model that allows shared randomness between the querying and interacting modules but no other information flow between them. We show that such $1$-round systems are efficiently emulated by standard testers whereas even $3/2$-round systems of extremely low complexity exist for properties that are extremely hard to test. We also show that in this model we can emulate some classes of IPPs that make proof-dependent queries.
Submitted to the Feinberg Graduate School of the Weizmann Institute of Science, Oct 2022.
Available: the thesis (in PDF file).
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