[Resource Topic] 2022/1407: Threshold Linear Secret Sharing to the Rescue of MPC-in-the-Head

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Threshold Linear Secret Sharing to the Rescue of MPC-in-the-Head

Authors: Thibauld Feneuil, Matthieu Rivain


The MPC-in-the-Head paradigm is a popular framework to build zero-knowledge proof systems using techniques from secure multi-party computation (MPC). While this paradigm is not restricted to a particular secret sharing scheme, all the efficient instantiations proposed so far rely on additive secret sharing.

In this work, we show how applying a threshold linear secret sharing scheme (threshold LSSS) can be beneficial to the MPC-in-the-Head paradigm. For a general MPC protocol model capturing most of the existing MPCitH schemes, we show that our approach improves the soundness of the underlying proof system from 1/N down to 1/\binom{N}{\ell}, where N is the number of parties and \ell is the threshold of the sharing scheme. While very general, our technique is limited to a number of parties N \leq |\mathbb{F}|, where \mathbb{F} is the field underlying the statement, because of the MDS conjecture.

Applying our approach with a low-threshold LSSS also boosts the performance of the proof system by making the MPC emulation cost independent of N for both the prover and the verifier. The gain is particularly significant for the verification time which becomes logarithmic in N (while the prover still has to generate and commit the N input shares). We further generalize and improve our framework: we show how homomorphic commitments can get rid of the linear complexity of the prover, we generalize our result to any quasi-threshold LSSS, and we describe an efficient batching technique relying on Shamir’s secret sharing.

We finally apply our techniques to specific use-cases. We first propose a variant of the recent SDitH signature scheme achieving new interesting trade-offs. In particular, for a signature size of 10 KB, we obtain a verification time lower than 0.5 ms, which is competitive with SPHINCS+, while achieving much faster signing. We further apply our batching technique to two different contexts: batched SDitH proofs and batched proofs for general arithmetic circuits based on the Limbo proof system. In both cases, we obtain an amortized proof size lower than 1/10 of the baseline scheme when batching a few dozen statements, while the amortized performances are also significantly improved.

ePrint: https://eprint.iacr.org/2022/1407

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