Compact All-but- \(\tau \) Vector Commitments for MPC-in-the-Head Signatures

Post-quantum signatures have become increasingly relevant as progress in quantum algorithms and active standardization efforts continue to shape the design space. Among the various post-quantum approaches, the MPC-in-the-Head paradigm offers a flexible way to obtain signatures by viewing them as non-interactive zero-knowledge proofs derived from multi-party computation. Vector commitments form a key part of this framework, and in many constructions the GGM-based all-but-one variant is responsible for a substantial fraction of the signing and verification cost. This work introduces a vector-commitment scheme that embeds the all-but- \(\tau \) semi-binding property into a single global GGM tree. Deriving all rounds within the same hierarchy removes the repeated opening steps associated with per-round structures, leaving the verifier with fewer authentication paths to process and reducing the cost of the commitment layer. To illustrate its applicability, we integrate the construction into the BN++ signature framework and obtain an optimized instantiation, denoted i-BN++. The integration does not alter the underlying MPC computation or the verification equations of BN++, and the security guarantees of the original design remain intact. The resulting construction shows that a global GGM tree can serve as a direct replacement for the per-round commitments traditionally used in MPCitH-based signatures.