**On Adaptive Security of Delayed-Input Sigma Protocols and Fiat-Shamir NIZKs**

*Michele Ciampi and Roberto Parisella and Daniele Venturi*

**Abstract: **We study adaptive security of delayed-input Sigma protocols and non-interactive zero-knowledge (NIZK) proof systems in the common reference string (CRS) model. Our contributions are threefold:

- We exhibit a generic compiler taking any delayed-input Sigma protocol and returning a delayed-input Sigma protocol satisfying adaptive-input special honest-verifier zero-knowledge (SHVZK). In case the initial Sigma protocol also satisfies adaptive-input special soundness, our compiler preserves this property. - We revisit the recent paradigm by Canetti et al. (STOC 2019) for obtaining NIZK proof systems in the CRS model via the Fiat-Shamir transform applied to so-called trapdoor Sigma protocols, in the context of adaptive security. In particular, assuming correlation-intractable hash functions for all sparse relations, we prove that Fiat- Shamir NIZKs satisfy either: (i) Adaptive soundness (and non-adaptive zero-knowledge), so long as the challenge is obtained by hashing both the prover’s first round and the instance being proven; (ii) Adaptive zero-knowledge (and non-adaptive soundness), so long as the challenge is obtained by hashing only the prover’s first round, and further assuming that the initial trapdoor Sigma protocol satisfies adaptive-input SHVZK. - We exhibit a generic compiler taking any Sigma protocol and returning a trapdoor Sigma protocol. Unfortunately, this transform does not preserve the delayed-input property of the initial Sigma protocol (if any). To complement this result, we also give yet another compiler taking any delayed-input trapdoor Sigma protocol and returning a delayed-input trapdoor Sigma protocol with adaptive-input SHVZK.

An attractive feature of our first two compilers is that they allow obtaining efficient delayed-input Sigma protocols with adaptive security, and efficient Fiat-Shamir NIZKs with adaptive soundness (and non-adaptive zero-knowledge) in the CRS model. Prior to our work, the latter was only possible using generic NP reductions.

**Category / Keywords: **foundations / Sigma protocols; Non-interactive zero knowledge; Adaptive security.

**Original Publication**** (with minor differences): **Conference on Security and Cryptography for Networks (SCN 2020)

**Date: **received 6 Jul 2020

**Contact author: **mciampi at ed ac uk,roberto@simula no,venturi@di uniroma1 it

**Available format(s): **PDF | BibTeX Citation

**Version: **20200707:084126 (All versions of this report)

**Short URL: **ia.cr/2020/831

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