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Consider the following $(2n+1)$ protocol:

$\mathcal{P}$ and $\mathcal{V}$ engage in an interaction where $\mathcal{P}$ consecutively sends a message $a_i$ answered by $\mathcal{V}$ with a random challenge $b_i$ for $i = 1,\dots,n$. Finally $\mathcal{P}$ gives a final answer $z$ and $\mathcal{V}$ outputs either $1$ or $0$ (i.e., accepts or rejects the proof) checking the conversation $(x,\{a_i\}_{i=1}^n,\{b_i\}_{i=1}^n,z)$.

The protocol verifies the following properties:

  • Completeness: If an honest prover $\mathcal{P}$ knows a valid witness $w$ and follows the protocol, then an honest verifier $\mathcal{V}$ always accepts the conversation.

  • $k$-Special Soundness: From $k$ valid conversations $\{(x,\{a_i^j\}_{i=1}^n,\{b_i^j\}_{i=1}^n,z^j)\}_{j=1}^k$, and $\{b_i^j\}_{i=1}^n \neq \{b_i^{j'}\}_{i=1}^n$ for all $j \neq j'$, it is possible to efficiently extract a witness $w$.

  • Honest-Verifier Zero-Knowledge: There exists a polynomial-time simulator that takes $x$ and random $\{b_i\}_{i=1}^n$ and output a valid conversation $(x,\{a_i\}_{i=1}^n,\{b_i\}_{i=1}^n,z)$ with the same probability distribution as conversations between honest $\mathcal{P}$ and $\mathcal{V}$.

This is a non-standard zero-knowledge protocol which I am trying to apply the Fiat-Shamir Heuristic to turn it into non-interactive. I am struggling with the soundness property, since is quite different from a typicial protocol.

Is there anything in the literature that generalizes the Fiat-Shamir Heuristic to finite $(2n+1)$ protocols? Maybe that can help to solve this problem.

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You can find the security proofs for 5-round Fiat-Shamir:

  • Ming-Shing Chen and Andreas Hülsing and Joost Rijneveld and Simona Samardjiska and Peter Schwabe: From 5-pass MQ-based identification to MQ-based signatures. Asiacrypt 2016. https://eprint.iacr.org/2016/708
  • Özgür Dagdelen, David Galindo, Pascal Véron, Sidi Mohamed El Yousfi Alaoui, and Pierre-Louis Cayrel. Extended security arguments for signature schemes. Designs, Codes and Cryptography, 78(2):441–461, 2016. https://link.springer.com/article/10.1007/s10623-014-0009-7

Their proof (may) contain (2n+1)-round Fiat-Shamir.

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  • $\begingroup$ The second paper did the trick $\endgroup$ – Lecter May 25 at 11:21

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