Efficient zero-knowledge proof for quadratic matrix relation over finite field with three or four witnesses

Abstract

In large-scale private computing applications, various arithmetic relations appear as or can be reduced to matrix relations. In this paper, we establish the efficient zero-knowledge proof (ZKP) for the quadratic matrix relation over finite field Fp with three or four witness matrices. In private computing tasks, lots of arithmetic relations are instances or special cases of such form, particularly some matrix structural decomposition relations. Different from the widely applied vectorspecific method, our method is matrix-specific. The matrix equation is treated as a tensor equality and probabilisticequivalent reduction techniques are applied to reduce the non-linear matrix relation to simple vector relation. To the authors’ best knowledge, currently, there are no matrix-specific methods to ZKP for nonlinear matrix relations. Compared against the current general linearization (vector-specific) method, our method substantially outperforms it in all critical aspects, e.g., for n-by-t matrix witnesses the required size of common reference string (c.r.s.) can be compressed by a factor of 2nt and the number of rounds, group and field elements in messages are all decreased by a factor of ≈2 for large-size witnesses. Computational complexities are almost the same in both methods.