# Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs

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## Abstract

A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry’s construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size \(|m|+{\mathrm {poly}}(k)\), where \(m\) is the plaintext and \(k\) is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.

## Keywords

Non-interactive zero-knowledge proofs Fully homomorphic encryption Hybrid encryption Secure function evaluation Minimizing communication## Notes

### Acknowledgments

Jens Groth was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 307937 and the Engineering and Physical Sciences Research Council Grant EP/G013829/1. Yuval Ishai was supported by the European Union’s Tenth Framework Programme (FP10/2010-2016) under Grant Agreement No. 259426 ERC-CaC, by ISF Grant 1361/10, and by BSF Grant 2012378. Chris Peikert was supported by the National Science Foundation under CAREER Award CCF-1054495, by the Alfred P. Sloan Foundation, and by the Defense Advanced Research Projects Agency (DARPA) and the Air Force Research Laboratory (AFRL) under Contract No. FA8750-11-C-0098. The views expressed are those of the authors and do not necessarily reflect the official policy or position of the National Science Foundation, the Sloan Foundation, DARPA or the U.S. Government. Amit Sahai was supported in part from a DARPA/ONR PROCEED award, NSFgrants 1228984, 1136174, 1118096, and 1065276, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through the U.S. Office of Naval Research under Contract N00014-11- 1-0389. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government. Adam Smith was supported by US National Science Foundation awards #0941553 and #0747294.

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