International Association
for Cryptologic Research

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): \begin{itemize} \item First, we present new statelessly updatable VCs with significantly shorter proofs than (and efficiency otherwise similar to) the only prior post-quantum, statelessly updatable construction (Papamanthou \etal, EUROCRYPT 13). Our constructions use private-key setup, in which an authority generates public parameters and then goes offline. \item Second, we construct functional commitments for \emph{arbitrary (bounded) Boolean circuits} and branching programs. Under falsifiable assumptions, this is the first post-quantum FC scheme beyond ordinary VCs, and the first FC scheme of any kind that goes beyond linearizable functions. Our construction works in a new model involving an authority that generates the public parameters and remains online to provide public, reusable ``opening keys'' for desired functions of committed messages. \end{itemize}

In recent years, there has been a proliferation of algebraically structured Learning With Errors (LWE) variants, including Ring-LWE, Module-LWE, Polynomial-LWE, Order-LWE, and Middle-Product LWE, and a web of reductions to support their hardness, both among these problems themselves and from related worst-case problems on structured lattices. However, these reductions are often difficult to interpret and use, due to the complexity of their parameters and analysis, and most especially their (frequently large) blowup and distortion of the error distributions.In this paper we unify and simplify this line of work. First, we give a general framework that encompasses all proposed LWE variants (over commutative base rings), and in particular unifies all prior “algebraic” LWE variants defined over number fields. We then use this framework to give much simpler, more general, and tighter reductions from Ring-LWE to other algebraic LWE variants, including Module-LWE, Order-LWE, and Middle-Product LWE. In particular, all of our reductions have easy-to-analyze and frequently small error expansion; in some cases they even leave the error unchanged. A main message of our work is that it is straightforward to use the hardness of the original Ring-LWE problem as a foundation for the hardness of all other algebraic LWE problems defined over number fields, via simple and rather tight reductions.