We present techniques for constructing zero-knowledge argument systems from garbled circuits, extending the GC-to-ZK compiler by Jawurek Et al. (2013) and the GC-to- \(\varSigma \) compiler by Hazay and Venkitasubramaniam (2020) to the following directions: − Our schemes are hybrid, commit-and-prove zero-knowledge argument systems that establish a connection between secrets embedded in algebraic commitments and a relation represented by a Boolean circuit. − Our schemes incorporate diverse cross-domain secrets embedded within distinct algebraic commitments, simultaneously supporting Pedersen-like commitments and lattice-based commitments. As an application, we develop circuit-represented compositions of \(\varSigma \) -protocols that support attractive access structures, such as weighted thresholds, that can be easily represented by a small circuit. For predicates \(P_1,\dots ,P_n\) individually associated with a \(\varSigma \) -protocol, and a predicate C represented by a Boolean circuit, we construct a \(\varSigma \) -protocol for proving \(C(P_1,\dots ,P_n)\) = 1. This result answers positively an open question posed by Abe, et. al. (2021).
