**Optimally Sound Sigma Protocols Under DCRA**

*Helger Lipmaa*

**Abstract: **Given a well-chosen additively homomorphic cryptosystem and a $\Sigma$ protocol with a linear answer, Damgård, Fazio, and Nicolosi proposed a non-interactive designated-verifier zero knowledge argument in the registered public key model that is sound under non-standard complexity-leveraging assumptions.
In 2015, Chaidos and Groth showed how to achieve the weaker yet reasonable culpable soundness notion under standard assumptions but only if the plaintext space order is prime. It makes use of $\Sigma$ protocols that satisfy what we call the \emph{optimal culpable soundness}. Unfortunately, most of the known additively homomorphic cryptosystems (like the Paillier Elgamal cryptosystem that is secure under the standard Decisional Composite Residuosity Assumption) have composite-order plaintext space. We construct optimally culpable sound $\Sigma$ protocols and thus culpably sound non-interactive designated-verifier zero knowledge protocols for NP under standard assumptions given that the least prime divisor of the plaintext space order is large.

**Category / Keywords: **public-key cryptography / Culpable soundness, designated verifier, homomorphic encryption, non-interactive zero knowledge, optimal soundness, registered public key model

**Date: **received 15 Jul 2017

**Contact author: **helger lipmaa at gmail com

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

**Version: **20170721:153731 (All versions of this report)

**Short URL: **ia.cr/2017/703

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