Short ZK‑PCPs

Determine whether there exist zero‑knowledge PCPs for NP with constant query complexity O(1) and nearly‑linear proof length \tilde{O}(n), i.e., construct perfect zero‑knowledge PCPs for NP whose prover output length matches quasilinear PCPs while retaining O(1) queries.

Background

The paper proves a zero‑knowledge PCP theorem: for any polynomial query bound q*, every NP language has a polynomial‑size, non‑adaptive perfect zero‑knowledge PCP with O(1) queries (via composition), matching the locality of the classical PCP theorem. However, the current construction’s proof length is polynomial rather than (quasi)linear.

Classical PCPs have been improved over decades to quasilinear proof length using Reed–Solomon arithmetization and combinatorial gap amplification, while the zero‑knowledge PCP approach here relies on Reed–Muller arithmetization and sumcheck‑based techniques. The authors explicitly pose whether similar near‑linear length can be achieved in the zero‑knowledge setting.