Recently, there has been a great development in communication-efficient zero-knowledge (ZK) protocols for arithmetic circuit relations. Since any relation can be translated into an arithmetic circuit relation, these primitives are extremely powerful and widely applied. However, this translation often comes at the cost of losing conceptual simplicity and modularity in cryptographic protocol design.For this reason, Lai et al. (CCS 2019), show how Bulletproof’s communication-efficient circuit zero-knowledge protocol (Bootle et al., EUROCRYPT 2016 and Bünz et al., S&P 2018) can be generalized to work for bilinear group arithmetic circuits directly, without requiring these circuits to be translated into arithmetic circuits. For many natural relations their approach is actually more efficient than the indirect circuit ZK approach.

We take a different approach and show that the arithmetic circuit model can be generalized to any circuit model in which (a) all wires take values in (possibly different) Zq-modules and (b) all gates have fan-in2and are either linear or bilinear mappings. We follow a straightforward generalization of Compressed Σ-Protocol Theory (CRYPTO 2020). We compress the communication complexity of a basic Σ-protocol for proving linear statements down to logarithmic. Then, we describe a linearization strategy to handle non-linearities. Besides its conceptual simplicity our approach also has practical advantages; we reduce the constant of the logarithmic component in the communication complexity of the CCS 2019 approach from 16 down to 6 and that of the linear component from 3 down to 1.

Moreover, the generalized commitment scheme required for bilinear circuit relations is also advantageous to standard arithmetic circuit ZK protocols, since its application immediately results in a square root reduction of public parameters size. The implications of this improvement can be significant, because many application scenarios result in very large sets of public parameters.

As an application of our compressed protocol for proving linear statements we construct the first k-out-of-n threshold signature scheme (TSS) with both transparent setup and threshold signatures of size O(κlog(n)) bits for security parameter κ. Each individual signature is of a so-called BLS type, the threshold signature hides the identities of the k signers and the threshold k can be dynamically chose n at aggregation time. Prior TSSs either result in sub-linear size signatures at the cost of requiring a trusted setup or the cost of the transparent setup amounts to linear (ink) size signatures.

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Attema, T, Cramer, R.J.F, & Rambaud, M. (2020). Compressed Sigma-Protocols for bilinear circuits and applications to logarithmic-sized transparent Threshold Signature Schemes.