Kronecker-FRI: A Hash-Based Multilinear Polynomial Commitment from a Coefficient-Extraction Identity

This work develops a Kronecker-product perspective on FRI-style polynomial proximity testing and coding-theoretic soundness. The central idea is to represent structured evaluation codes through tensor products and to analyze folding, distance, and decoding properties directly in this Kronecker framework. This formulation makes the recursive structure underlying FRI explicit and provides a convenient algebraic language for studying high-dimensional codewords, multilinear structure, and proximity to tensor-product codes. The document develops the mathematical foundations needed for this viewpoint, including Kronecker products of linear codes, nonzero conditions for tensor codewords, distance properties, polynomial evaluation structure, and decoding arguments. It also explains how these ingredients connect to FRI-style recursive reductions and post-quantum proof systems based on hash commitments and Reed–Solomon-type proximity testing. The intended applications are in zero-knowledge proofs, post-quantum polynomial commitment schemes, proof-system design, and the analysis of FRI-based protocols. Keywords: FRI, Fast Reed–Solomon Interactive Oracle Proofs of Proximity, Kronecker product, tensor codes, Reed–Solomon codes, polynomial commitments, multilinear polynomials, proximity testing, error-correcting codes, zero-knowledge proofs, post-quantum cryptography.