Paper 2026/1571 Original: eprint.iacr.org/2026/1571 ↗

LAMP: Linear Verification of Matrix Multiplication via Proximity Testing

Abstract

Verifiable computation systems often need to prove large matrix multiplication statements, but a direct SNARK arithmetization of a \(k \times k\) product requires \(\mathcal{O}(k^3)\) constraints. Freivalds' randomized check reduces the algebraic computation to vector-matrix products, but proving those products inside a SNARK still costs \(\mathcal{O}(k^2)\) constraints. We present LAMP, a matrix-multiplication checking protocol that combines Freivalds' randomized check with proximity testing over linear error-correcting codes. The prover commits to encoded matrices and intermediate vectors before the sampled query positions are derived. The CP-SNARK circuit then checks only the sampled codeword positions and commits to the values used inside the circuit, while Merkle openings and CP-Link proofs ensure consistency between the in-circuit witnesses and the externally committed values. We prove soundness for this committed-input setting under the soundness of the SNARK backend, the binding of the commitments, the correctness of the CP-Link checks, and the relative distance and unique-decoding radius of the Reed--Solomon code. For a fixed number \(t\) of sampled positions, the main in-circuit SNARK relation has \(\mathcal{O}(tk+E_{\mathsf{code}})\) constraints, with additional \(\mathcal{O}(t\log n)+E_{\mathsf{link}}(k,t)\) backend work for Merkle openings and CP-Link checks. We implement LAMP in Go and compare it with a Freivalds-based SNARK circuit. In the matrix benchmark at \(k=2^{12}\), LAMP reduces the constraint count by \(9.68\times\) and shortens proof generation time by \(5.81\times\); verification remains approximately \(0.13\,\text{s}\) across the measured range.