Abstract
For many applications of secure multiparty computation it is natural to demand that the output of the protocol is verifiable. Verifiability should ensure that incorrect outputs are always rejected, even if all parties executing the secure computation collude. Since the inputs to a secure computation are private, and potentially the outputs are private as well, adding verifiability is in general hard and costly.
In this paper we focus on privacy-preserving linear programming as a typical and practically relevant case for verifiable secure multiparty computation. We introduce certificate validation as an effective technique for achieving verifiable linear programming. Rather than verifying the computation proper, which involves many iterations of the simplex algorithm, we extend the output of the secure computation with a certificate. The certificate allows for efficient and direct validation of the correctness of the output. The overhead incurred by the computation of the certificate is marginal. For the validation of a certificate we design particularly efficient distributed-prover zero-knowledge proofs, fully exploiting the fact that we can use ElGamal encryption for this purpose, hence avoiding the use of more elaborate cryptosystems such as Paillier encryption.
We also formulate appropriate security definitions for our approach, and prove security for our protocols in this model, paying special attention to ensuring properties such as input independence. By means of several experiments performed in a real multi-cloud-provider environment, we show that the overall performance for verifiable linear programming is very competitive, incurring minimal overhead compared to protocols providing no correctness guarantees at all.
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Notes
- 1.
Of course, parties can alternatively share \(a^{-s_1}\), \(a^{-s_2}\); we prefer our description because it treats decryption and decryption shares uniformly.
- 2.
Available at http://www.win.tue.nl/~berry/TUeVIFF/.
- 3.
http://www.netlib.org/lp/data/; coefficients rounded for performance.
- 4.
We took the minimal bitlengths needed for correctness. In practice, these are not known in advance: for VIFF, one takes a safe margin; for the proof, one can reveal and use the maximal bit length of all bit decompositions in the certificate.
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Acknowledgements
We thank Dan Bernstein, Thijs Laarhoven, Peter Nordholt, and Niels de Vreede for useful discussions, and the anonymous reviewers for their suggestions. This work was supported in part by the European Commission through the ICT program under contract INFSO-ICT-284833 (PUFFIN); through the FP7 programme under grant 609611 (PRACTICE); and through the H2020 programme under grant 643964 (SUPERCLOUD).
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de Hoogh, S., Schoenmakers, B., Veeningen, M. (2016). Certificate Validation in Secure Computation and Its Use in Verifiable Linear Programming. In: Pointcheval, D., Nitaj, A., Rachidi, T. (eds) Progress in Cryptology – AFRICACRYPT 2016. AFRICACRYPT 2016. Lecture Notes in Computer Science(), vol 9646. Springer, Cham. https://doi.org/10.1007/978-3-319-31517-1_14
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