Paper 2026/1196 Original: eprint.iacr.org/2026/1196 ↗

Grand Danois: Succinct Multilinear Polynomial Commitments over Lattices

Abstract

We present Grand Danois, a new post-quantum multilinear polynomial commitment scheme from lattices for polynomials over $\mathbb{F}_q$ that achieves polylogarithmic $O(\lambda \ell)$ verification complexity and proof sizes. We build on the general approach introduced in Hachi (ePrint 2026/156) with three key changes. First, we switch to the vanishing Short Integer Solution (vSIS) assumption to obtain structured public parameters for our commitment scheme and utilize this structure to design a sumcheck protocol amenable to succinct verification. Second, rather than casting ring relations into $\mathbb{F}_{q^k}[X]$, via the residual technique of Hachi, we express multiplication by fixed $\mathcal{R}_q$ elements through its rotation matrix, which lets the verifier fold each row of the constraint matrix in time linear rather than quadratic in the ring degree $d$. Third, we modify the quadratic relation used in Hachi and Greyhound (CRYPTO 2024) so that it becomes compatible with proving norm bounds using Johnson-Lindenstrauss projections. This is achieved through an adaptation of the structured projection strategy introduced in RoK and Roll (ASIACRYPT 2025). This has the benefit for communication complexity in that proving norm bounds and correct polynomial evaluation are integrated into a single protocol, reducing the number of commitments sent by the prover. Furthermore, we impose additional structure on our random projections to reduce the witness size even more aggressively during each round of recursion without sacrificing verification complexity. Under the vSIS assumption, our construction yields an estimated proof size of roughly $80$ KB for $2^{32}$-size polynomial evaluations.