Dynamic vector commitments that enable local updates of opening proofs have applications ranging from verifiable databases with membership changes to stateless clients on blockchains. In these applications, each user maintains a relevant subset of the committed messages and the corresponding opening proofs with the goal of ensuring a succinct global state. When the messages are updated, users are given some global update information and update their opening proofs to match the new vector commitment. We investigate the relation between the size of the update information and the runtime complexity needed to update an individual opening proof. Existing vector commitment schemes require that either the information size or the runtime scale linearly in the number k of updated state elements. We construct a vector commitment scheme that asymptotically achieves both length and runtime that is sublinear in k. We prove an information-theoretic lower bound on the relation between the update information size and runtime complexity that shows the asymptotic optimality of our scheme. While in practice, the construction is not yet competitive with Verkle commitments, our approach may point the way towards more performant vector commitments.

翻译：动态向量承诺能够实现开放证明的局部更新，其应用涵盖支持成员变更的可验证数据库以及区块链上的无状态客户端。在这些应用中，每个用户维护已承诺消息的相关子集及对应的开放证明，目标是确保简洁的全局状态。当消息被更新时，用户获取全局更新信息并更新其开放证明以匹配新的向量承诺。我们研究了更新信息大小与更新单个开放证明所需运行时复杂度之间的关系。现有向量承诺方案要求信息大小或运行时与更新状态元素数量k呈线性关系。我们构建了一种渐近实现信息长度与运行时均关于k为次线性的向量承诺方案。我们证明了更新信息大小与运行时复杂度之间信息论下界的渐近最优性。尽管在实际应用中该构造尚不如Verkle承诺具有竞争力，但我们的方法可能为构建更高效的向量承诺指明方向。