Efficient zero-knowledge proof for quadratic matrix relation over finite field with two 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 two witness matrices. In private computing tasks, lots of typical relations are instances or special cases of this form, e.g., matrix multiplicative relation, inverse relation, isometric relation, etc. 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 (vector-specific) method, our method substantially outperforms it in all critical aspects, e.g., for n-raw t-column matrix witnesses, the required size of common reference string (c.r.s.) can be compressed by a factor of 2nt and the number of messages, group and field elements are all reduced by a factor of ≈2 for large-size witnesses. Computational complexities in both methods are almost the same.