Designated-Verifier Dynamic zk-SNARKs with Applications to Dynamic Proofs of Index

Recently, the notion of dynamic zk-SNARKs was introduced. A dynamic zk-SNARK augments a standard zk-SNARK with an efficient update algorithm. Given a valid source statement-witness pair (𝑥, 𝑤) together with a verifying proof 𝑝, and a valid target statement-witness pair (𝑥 ′ , 𝑤 ′ ), the update algorithm outputs a verifying proof 𝑝 ′ for (𝑥 ′ , 𝑤 ′ ). Crucially, 𝑝 ′ is not recomputed from scratch; instead, the update algorithm takes time roughly proportional to the Hamming distance between (𝑥, 𝑤) and (𝑥 ′ , 𝑤 ′ ), analogous to how dynamic data structures update the result of a computation after a small change. In this paper, we initiate the study of designated-verifier dynamic zk-SNARKs: dynamic zk-SNARKs in which only a designated verifier, holding secret verification state, can be convinced by a proof. Following recent advances in designated-verifier zk-SNARKs-such as efficient post-quantum designated verifier SNARKs (CCS 2021) and designated verifier SNARKs with very small proofs (CRYPTO 2025)-we construct a designated-verifier dynamic zk-SNARK with 𝑂 (log 𝑛) update time, constant proof size, and concrete efficiency. Our construction significantly outperforms Dynalog (both asymptotically and concretely), the only publicly verifiable dynamic zk-SNARK with polylogarithmic update time (Wang et al., 2024) . The concrete efficiency of our construction enables, for the first time, an efficient implementation of a dynamic proof of index: Given a digest 𝑑 of an arbitrary set and a digest 𝑑 ′ of its sorted index (e.g., binary search tree), we produce a SNARK proof certifying the consistency of 𝑑 and 𝑑 ′ . More importantly, this proof can be updated in sublinear time when the underlying set changes-for example, when an element is modified or inserted, potentially altering the sorted order. We demonstrate applications of designated-verifier dynamic proofs of index to verifiable dynamic database outsourcing, where a client outsources a database and later maintains verifiable indices for efficient query answering, even under arbitrary database updates. Our main contribution is Delphus, an efficient designated-verifier dynamic zk-SNARK. To the best of our knowledge, Delphus has