Abstract
Homomorphic universally composable (UC) commitments allow for the sender to reveal the result of additions and multiplications of values contained in commitments without revealing the values themselves while assuring the receiver of the correctness of such computation on committed values. In this work, we construct essentially optimal additively homomorphic UC commitments from any (not necessarily UC or homomorphic) extractable commitment, while the previous best constructions require oblivious transfer. We obtain amortized linear computational complexity in the length of the input messages and rate 1. Next, we show how to extend our scheme to also obtain multiplicative homomorphism at the cost of asymptotic optimality but retaining low concrete complexity for practical parameters. Moreover, our techniques yield public coin protocols, which are compatible with the Fiat-Shamir heuristic. These results come at the cost of realizing a restricted version of the homomorphic commitment functionality where the sender is allowed to perform any number of commitments and operations on committed messages but is only allowed to perform a single batch opening of a number of commitments. Although this functionality seems restrictive, we show that it can be used as a building block for more efficient instantiations of recent protocols for secure multiparty computation and zero knowledge non-interactive arguments of knowledge.
I. Cascudo—This work was done while Ignacio Cascudo was with Department of Mathematics, Aalborg University, Denmark.
I. Damgård and R. Dowsley—This project has received funding from the European Research Council (ERC) under the European Unions’ Horizon 2020 research and innovation programme under grant agreement No 669255 (MPCPRO).
B. David—This work was partially supported by DFF grant number 9040-00399B (TrA\(^{2}\)C).
I. Giacomelli—This work was done while Irene Giacomelli was with the ISI Foundation (Turin, Italy) and supported by Intesa Sanapolo Innovation Center.
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Notes
- 1.
All this holds in an amortized sense, assuming we make enough commitments so that the cost of the setup phase is dwarfed.
- 2.
The scheme of [9] can be constructed from an extractable commitment and an equivocal commitment. However, it is intrinsically incompatible with homomorphic operations.
- 3.
This argument works, even if the prover did not choose a codeword at commit time. If we also want to have additive homomorphism, we need to check that the prover chose something that it at least close to a codeword. This can be done using, e.g., the interactive proximity testing from [16].
- 4.
Recall that erasure correction for linear codes can be performed efficiently via gaussian elimination.
- 5.
More precisely, the last share of each coordinate of \(\mathsf {C}(\mathbf {y})\) is added with the corresponding (now public) coordinate of \(\mathsf {C}(\mathbf {a}*\mathbf {b}-\mathbf {y})\).
- 6.
One may think that the tighter lower bound \(\mathsf {dist}(\widetilde{\mathsf {C}})\ge \mathsf {dist}(\mathsf {C})+\mathsf {dist}(\mathsf {C}^{*2})\) holds, but this is not necessarily true if the dimension \(k'\) of \(\mathsf {C}^{*2}\) is larger than k, as in that case there will be codewords of the form \((\mathbf {0^k},\mathbf {0^{n-k}},\mathbf {0^k},\mathbf {c'})\) where \(\mathbf {c'}\ne \mathbf {0^{n-k}}\). Indeed take \((\mathbf {0^k},\mathbf {c'})\) to be the encoding by \(\mathsf {C}^{*2}\) of \((\mathbf {0^k}||\mathbf {z})\) for a nonzero \(\mathbf {z}\in \{0,1\}^{k'-k}\).
- 7.
And naturally from this one can also obtain a \([4095\cdot \ell ,338\cdot \ell ]\)-code with the same minimum distance of its square, by simply applying the [4095, 338] to each block of 338 bits of the message.
- 8.
In [31] they are referred to as multi-commitments.
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Cascudo, I., Damgård, I., David, B., Döttling, N., Dowsley, R., Giacomelli, I. (2019). Efficient UC Commitment Extension with Homomorphism for Free (and Applications). In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11922. Springer, Cham. https://doi.org/10.1007/978-3-030-34621-8_22
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