International Association for Cryptologic Research

International Association
for Cryptologic Research


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

Thibauld Feneuil , CryptoExperts, Sorbonne Université, France
Matthieu Rivain , CryptoExperts, France
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Presentation: Slides
Conference: ASIACRYPT 2023
Abstract: 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 for small circuits 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 passively-secure 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/binomial(N,\ell), where N is the number of parties and \ell is the privacy threshold of the sharing scheme. While very general, our technique is limited to a number of parties N <= |\F|, where \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 linearly-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.
  title={Threshold Linear Secret Sharing to the Rescue of MPC-in-the-Head},
  author={Thibauld Feneuil and Matthieu Rivain},