Sub-Linear Lattice-Based Zero-Knowledge Arguments for Arithmetic Circuits

We propose the first zero-knowledge argument with sub-linear communication complexity for arithmetic circuit satisfiability over a prime $p$ whose security is based on the hardness of the short integer solution (SIS) problem. For a circuit with $N$ gates, the communication complexity of our protocol is $O\left(\sqrt{N\lambda\log^3{N}}\right)$, where $\lambda$ is the security parameter. A key component of our construction is a surprisingly simple zero-knowledge proof for pre-images of linear relations whose amortized communication complexity depends only logarithmically on the number of relations being proved. This latter protocol is a substantial improvement, both theoretically and in practice, over the previous results in this line of research of Damgård et al. (CRYPTO 2012), Baum et al. (CRYPTO 2016), Cramer et al. (EUROCRYPT 2017) and del Pino and Lyubashevsky (CRYPTO 2017), and we believe it to be of independent interest