Practical Exact Proofs from Lattices: New Techniques to Exploit Fully-Splitting Rings

Abstract

We propose a very fast lattice-based zero-knowledge proof system for exactly proving knowledge of a ternary solution $\vec{s} \in \{-1,0,1\}^n$ to a linear equation $A\vec{s}=\vec{u}$ over $\mathbb{Z}_q$, which improves upon the protocol by Bootle, Lyubashevsky and Seiler (CRYPTO 2019) by producing proofs that are shorter by a factor of $8$. At the core lies a technique that utilizes the module-homomorphic BDLOP commitment scheme (SCN 2018) over the fully splitting cyclotomic ring $\mathbb{Z}_q[X]/(X^d + 1)$ to prove scalar products with the NTT vector of a secret polynomial