## Public-Coin Time- and Space-Efficient Arguments

### Justin Holmgren

**Abstract:**

We construct public-coin time- and space-efficient zero-knowledge arguments for NP where, for any time-T, space-S non-deterministic RAM computation, the prover runs in time T * polylog(T) and space S * polylog(T), and the verifier runs in time n * polylog(T), where n is the input length of the computation. Our protocol relies on groups whose order is not efficiently computable, which:

can be instantiated with a trusted setup from the hardness of factoring products of safe primes, can be instantiated without a trusted setup, using class groups.

Our argument-system can heuristically be made non-interactive using the Fiat-Shamir transform.

Our proof builds on DARK (Bünz et al., Eurocrypt 2020), a recently proposed polynomial commitment scheme. We show how to implement a variant of DARK in a time- and space-efficient way. Along the way we also:

1. Identify a significant gap in the proof of security of DARK. 2. Non-trivially modify the DARK scheme to overcome the aforementioned gap. The modified version relies on weaker cryptographic assumptions than those in the original DARK scheme. Our proof utilizes ideas from the theory of integer lattices in a novel way. 3. Generalize Pietrzak's (ITCS 2019) proof of exponentiation (PoE) protocol to work with general groups of unknown order (without relying on any cryptographic assumption).

In proving these results, we develop general-purpose techniques for working with (hidden order) groups, which may be of independent interest.

Based on joint work with Alexander Block, Alon Rosen, Ron Rothblum, and Pratik Soni.