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Abstract Zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) are cryptographic protocols that offer efficient and privacy-preserving means of verifying NP language relations and have drawn considerable attention for their appealing applications, e.g., verifiable computation and anonymous payment protocol. Compared with the pre-quantum case, the practicability of this primitive in the post-quantum setting is still unsatisfactory, especially for the space complexity. To tackle this issue, this work seeks to enhance the efficiency and compactness of lattice-based zk-SNARKs, including proof length and common reference string (CRS) length. In this paper, we develop the framework of square span program-based SNARKs and design new zk-SNARKs over cyclotomic rings. Compared with previous works, our construction is without parallel repetition and achieves shorter proof and CRS lengths than previous lattice-based zk-SNARK schemes. Particularly, the proof length of our scheme is around$$23.3\%$$ $23.3\%$smaller than the recent shortest lattice-based zk-SNARKs by Ishai et al. (in: Proceedings of the 2021 ACM SIGSAC conference on computer and communications security, pp 212–234, 2021), and the CRS length is$$3.6\times$$ $3.6\times$smaller. Our constructions follow the framework of Gennaro et al. (in: Proceedings of the 2018 ACM SIGSAC conference on computer and communications security, pp 556–573, 2018), and adapt it to the ring setting by slightly modifying the knowledge assumptions. We develop concretely small constructions by using module-switching and key-switching procedures in a novel way. 

In this work, we conduct a comprehensive study on establishing hardness reductions for (Module) Learning with Rounding over rings (RLWR). Towards this, we present an algebraic framework of LWR, inspired by a recent work of Peikert and Pepin (TCC ’19). Then we show a search-to-decision reduction for Ring-LWR, generalizing a result in the plain LWR setting by Bogdanov et al. (TCC ’15). Finally, we show a reduction from Ring-LWE to Module Ring-LWR (even for leaky secrets), generalizing the plain LWE to LWR reduction by Alwen et al. (Crypto ’13). One of our central techniques is a new ring leftover hash lemma, which might be of independent interests.