Constant-Round Arguments for Batch-Verification and Bounded-Space Computations from One-Way Functions

What are the minimal cryptographic assumptions that suffice for constructing efficient argument systems, and for which tasks? Recently, Amit and Rothblum [STOC 2023] showed that one-way functions suffice for constructing constant-round arguments for bounded-depth computations. In this work we ask: what other tasks have efficient argument systems based only on one-way functions? We show two positive results: First, we construct a new argument system for batch-verification of k \(\textsf{UP}\) statements ( \(\textsf{NP}\) statements with a unique witness) for witness relations that are verifiable in depth D. Taking M to be the length of a single witness, the communication complexity is \(O(\log k) \cdot (M + k \cdot D \cdot n^{\sigma })\) , where \(\sigma > 0\) is an arbitrarily small constant. In particular, the communication is quasi-linear in the length of a single witness, so long as \({k < M / (D \cdot n^{\sigma })}\) . The number of rounds is constant and the honest prover runs in polynomial time given witnesses for all k inputs’ membership in the language. Our second result is a constant-round doubly-efficient argument system for languages in \(\textsf{P}\) that are computable by bounded-space Turing machines. For this class of computations, we obtain an exponential improvement in the trade-off between the number of rounds and the (exponent of the) communication complexity, compared to known unconditionally sound protocols [Reingold, Rothblum and Rothblum, STOC 2016].