A Sub-linear Lattice-Based Submatrix Commitment Scheme

Abstract

Subvector commitment is a recently proposed cryptographic primitive that provides the underlying cryptographic tool to design many interesting security systems such as succinct non-interactive arguments of knowledge (SNARK), verifiable database, dynamic accumulators, etc. In this paper, we present a generalization of subvector commitment, a public-coin-setup lattice-based submatrix commitment, which allows a commitment of a message matrix to be opened on multiple entries of the matrix simultaneously. It exploits a conceptual similarity between Single-Instruction Multiple-Data (SIMD) in homomorphic encryption and submatrix commitment, and develops a novel position binding technique based on the Chinese Remainder Theorem. We show that the position binding property can be reduced to module-based short integer solution (SIS) problem, a standard assumption that is believed to be post-quantum secure. We also show that the commitment and opening size of our commitment scheme are both sublinear, i.e., proportional to the square root of the message size. As far as we know, this is the first public-coin-setup and post-quantum secure subvector commitment scheme.