Abstract
We consider the problem of Non-Interactive Two-Party Secure Computation (NISC), where Rachel wishes to publish an encryption of her input x, in such a way that any other party, who holds an input y, can send her a single message which conveys to her the value f(x, y), and nothing more. We demand security against malicious parties. While such protocols are easy to construct using garbled circuits and general non-interactive zero-knowledge proofs, this approach inherently makes a non-black-box use of the underlying cryptographic primitives and is infeasible in practice.
Ishai et al. (Eurocrypt 2011) showed how to construct NISC protocols that only use parallel calls to an ideal oblivious transfer (OT) oracle, and additionally make only a black-box use of any pseudorandom generator. Combined with the efficient 2-message OT protocol of Peikert et al. (Crypto 2008), this leads to a practical approach to NISC that has been implemented in subsequent works. However, a major limitation of all known OT-based NISC protocols is that they are subject to selective failure attacks that allows a malicious sender to entirely compromise the security of the protocol when the receiver’s first message is reused.
Motivated by the failure of the OT-based approach, we consider the problem of basing reusable NISC on parallel invocations of a standard arithmetic generalization of OT known as oblivious linear-function evaluation (OLE). We obtain the following results:
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We construct an information-theoretically secure reusable NISC protocol for arithmetic branching programs and general zero-knowledge functionalities in the OLE-hybrid model. Our zero-knowledge protocol only makes an absolute constant number of OLE calls per gate in an arithmetic circuit whose satisfiability is being proved. We also get reusable NISC in the OLE-hybrid model for general Boolean circuits using any one-way function.
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We complement this by a negative result, showing that reusable NISC is impossible to achieve in the OT-hybrid model. This provides a formal justification for the need to replace OT by OLE.
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We build a universally composable 2-message reusable OLE protocol in the CRS model that can be based on the security of Paillier encryption and requires only a constant number of modular exponentiations. This provides the first arithmetic analogue of the 2-message OT protocols of Peikert et al. (Crypto 2008).
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By combining our NISC protocol in the OLE-hybrid model and the 2-message OLE protocol, we get protocols with new attractive asymptotic and concrete efficiency features. In particular, we get the first (designated-verifier) NIZK protocols for NP where following a statement-independent preprocessing, both proving and verifying are entirely “non-cryptographic” and involve only a constant computational overhead. Furthermore, we get the first statistical designated-verifier NIZK argument for NP under an assumption related to factoring.
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Notes
- 1.
Alternatively, settling for computational security, one can replace a field by a “pseudo-field,” namely a ring whose description hides all zero-divisors. A useful example is the ring \(\mathbb {Z}_N\) for an RSA modulus N with an unknown factorization.
- 2.
The actual roadmap is somewhat different, and will be gradually revealed in this overview. An impatient reader is referred to Fig. 1 at the end of the overview.
- 3.
In the main body, we do not need to assume that y is random. Moreover, we will consider more general arithmetic conditions beyond \(z=xy\).
- 4.
This is because the first component of the ciphertext, \(b^r\) contains no information about \(r\mod N\).
- 5.
There is a minor subtlety here, where because \(G_{2p'q'}\) has an order 2 subgroup an extra component in this subgroup might not be detected; to prevent this, we actually square all the elements during decryption to eliminate this subgroup, and then decrypt the final result divided by 2.
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Acknowledgements
Y. Dodis was supported in part by gifts from VMware Labs, Facebook and Google, and NSF grants 1314568, 1619158, 1815546. Y. Ishai was supported by ERC Project NTSC (742754), ISF grant 1709/14, NSF-BSF grant 2015782, and a grant from the Ministry of Science and Technology, Israel and Department of Science and Technology, Government of India. D. Kraschewski Supported by the European Union’s Tenth Framework Programme (FP10/2010-2016) under grant agreement no. 259426 - ERC Cryptography and Complexity. Work mostly done while at the Technion. T. Liu was supported in part by NSF Grants CNS-1350619, CNS-1414119 and CNS-1718161, an MIT-IBM grant and a DARPA Young Faculty Award. R. Ostrovsky was supported by NSF grant 1619348, BSF grant 2015782, DARPA SafeWare subcontract to Galois Inc., DARPA SPAWAR contract N66001-15-C-4065, JP Morgan Faculty Research Award, OKAWA Foundation Research Award, IBM Faculty Research Award, Xerox Faculty Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. The views expressed are those of the authors and do not reflect position of the Department of Defense or the U.S. Government. V. Vaikuntanathan was supported in part by NSF Grants CNS-1350619, CNS-1414119 and CNS-1718161, an MIT-IBM grant and a DARPA Young Faculty Award.
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Chase, M. et al. (2019). Reusable Non-Interactive Secure Computation. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11694. Springer, Cham. https://doi.org/10.1007/978-3-030-26954-8_15
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