Vector and Functional Commitments from Lattices

Vector commitment (VC) schemes allow one to commit concisely to an ordered sequence of values, so that the values at desired positions can later be proved concisely. In addition, a VC can be statelessly updatable, meaning that commitments and proofs can be updated to reflect changes to individual entries, using knowledge of just those changes (and not the entire vector). VCs have found important applications in verifiable outsourced databases, cryptographic accumulators, and cryptocurrencies. However, to date there have been relatively few post-quantum constructions, i.e., ones that are plausibly secure against quantum attacks.More generally, functional commitment (FC) schemes allow one to concisely and verifiably reveal various functions of committed data, such as linear functions (i.e., inner products, including evaluations of a committed polynomial). Under falsifiable assumptions, all known functional commitments schemes have been limited to “linearizable” functions, and there are no known post-quantum FC schemes beyond ordinary VCs.In this work we give post-quantum constructions of vector and functional commitments based on the standard Short Integer Solution lattice problem (appropriately parameterized):