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**Secure arithmetic computation with no honest majority.**
*(English)*
Zbl 1213.94111

Reingold, Omer (ed.), Theory of cryptography. 6th theory of cryptography conference, TCC 2009, San Francisco, CA, USA, March 15–17, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-00456-8/pbk). Lecture Notes in Computer Science 5444, 294-314 (2009).

Summary: We study the complexity of securely evaluating arithmetic circuits over finite rings. This question is motivated by natural secure computation tasks. Focusing mainly on the case of two-party protocols with security against malicious parties, our main goals are to: (1) only make black-box calls to the ring operations and standard cryptographic primitives, and (2) minimize the number of such black-box calls as well as the communication overhead.

We present several solutions which differ in their efficiency, generality, and underlying intractability assumptions. These include:

– An unconditionally secure protocol in the OT-hybrid model which makes a black-box use of an arbitrary ring \(R\),but where the number of ring operations grows linearly with (an upper bound on) \(\log |R|\).

– Computationally secure protocols in the OT-hybrid model which make a black-box use of an underlying ring, and in which the number of ring operations does not grow with the ring size. The protocols rely on variants of previous intractability assumptions related to linear codes. In the most efficient instance of these protocols, applied to a suitable class of fields, the (amortized) communication cost is a constant number of field elements per multiplication gate and the computational cost is dominated by \(O(\log k)\) field operations per gate, where \(k\) is a security parameter. These results extend a previous approach of M. Naor and B. Pinkas for secure polynomial evaluation [“Oblivious polynomial evaluation”, SIAM J. Comput. 35, No. 5, 1254–1281 (2006; Zbl 1101.94026)].

– A protocol for the rings \(\mathbb Z_{m } = \mathbb Z/m\mathbb Z\) which only makes a black-box use of a homomorphic encryption scheme. When \(m\) is prime, the (amortized) number of calls to the encryption scheme for each gate of the circuit is constant. All of our protocols are in fact UC-secure in the OT-hybrid model and can be generalized to multiparty computation with an arbitrary number of malicious parties.

For the entire collection see [Zbl 1156.94005].

We present several solutions which differ in their efficiency, generality, and underlying intractability assumptions. These include:

– An unconditionally secure protocol in the OT-hybrid model which makes a black-box use of an arbitrary ring \(R\),but where the number of ring operations grows linearly with (an upper bound on) \(\log |R|\).

– Computationally secure protocols in the OT-hybrid model which make a black-box use of an underlying ring, and in which the number of ring operations does not grow with the ring size. The protocols rely on variants of previous intractability assumptions related to linear codes. In the most efficient instance of these protocols, applied to a suitable class of fields, the (amortized) communication cost is a constant number of field elements per multiplication gate and the computational cost is dominated by \(O(\log k)\) field operations per gate, where \(k\) is a security parameter. These results extend a previous approach of M. Naor and B. Pinkas for secure polynomial evaluation [“Oblivious polynomial evaluation”, SIAM J. Comput. 35, No. 5, 1254–1281 (2006; Zbl 1101.94026)].

– A protocol for the rings \(\mathbb Z_{m } = \mathbb Z/m\mathbb Z\) which only makes a black-box use of a homomorphic encryption scheme. When \(m\) is prime, the (amortized) number of calls to the encryption scheme for each gate of the circuit is constant. All of our protocols are in fact UC-secure in the OT-hybrid model and can be generalized to multiparty computation with an arbitrary number of malicious parties.

For the entire collection see [Zbl 1156.94005].

### MSC:

94A60 | Cryptography |

68Q25 | Analysis of algorithms and problem complexity |

94A62 | Authentication, digital signatures and secret sharing |