Vector commitments (VCs), enabling to commit to a vector and locally reveal any of its entries, play a key role in a variety of both classic and recently evolving applications. However, security notions for VCs have so far focused on passive attacks, and non-malleability notions considering active attacks have not been explored. Moreover, existing frameworks that may enable to capture the non-malleability of VCs seem either too weak (non-malleable non-interactive commitments that do not account for the security implications of local openings) or too strong (non-malleable zero-knowledge sets that support both membership and non-membership proofs). We put forward a rigorous framework capturing the non-malleability of VCs, striking a careful balance between the existing weaker and stronger frameworks: We strengthen the framework of non-malleable non-interactive commitments by considering attackers that may be exposed to local openings, and we relax the framework of non-malleable zero-knowledge sets by focusing on membership proofs. In addition, we strengthen both frameworks by supporting (inherently private) updates to entries of committed vectors, and discuss the benefits of non-malleable VCs in the context of both UTXO-based and account-based stateless blockchains, and in the context of simultaneous multi-round auctions (that have been adopted by the US Federal Communications Commission as the standard auction format for selling spectrum ranges). Within our framework, we present a direct approach for constructing non-malleable VCs whose efficiency essentially matches that of the existing standard VCs. Specifically, we show that any VC can be transformed into a non-malleable one, relying on a new primitive that we put forth. Our new primitive, locally equivocable commitments with all-but-one binding, is evidently both conceptually and technically simpler compared to multi-trapdoor mercurial trapdoor commitments (the main building block underlying existing non-malleable zero-knowledge sets), and admits more efficient instantiations based on the same number-theoretic assumptions.
Bibliographical noteFunding Information:
Supported by the Israel Science Foundation (Grant No. 1336/22) and by the European Union (ERC, FTRC, 101043243). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
© 2023, International Association for Cryptologic Research.
- Vector commitments