Practical Product Proofs for Lattice Commitments

We construct a practical lattice-based zero-knowledge argument for proving multiplicative relations between committed values. The underlying commitment scheme that we use is the currently most efficient one of Baum et al. (SCN 2018), and the size of our multiplicative proof (9 KB) is only slightly larger than the 7 KB required for just proving knowledge of the committed values. We additionally expand on the work of Lyubashevsky and Seiler (Eurocrypt 2018) by showing that the above-mentioned result can also apply when working over rings ℤ𝑞[𝑋]/(𝑋𝑑+1) where 𝑋𝑑+1 splits into low-degree factors, which is a desirable property for many applications (e.g. range proofs, multiplications over ℤ𝑞 ) that take advantage of packing multiple integers into the NTT coefficients of the committed polynomial.