🤖 AI Summary
This work addresses the lack of a composability theory for arithmetic masking over prime fields, which undermines side-channel security guarantees for NTT-based post-quantum cryptographic hardware in multi-stage pipelines. The authors propose a Prime-Field Probing-Isolating Non-Interference (PF-PINI) model that introduces an inter-stage “mask refreshing” mechanism to fully uniformize intermediate values, thereby enabling secure composition. They establish the first machine-verified composability theorem for prime-field arithmetic masking, formally proving that mask refreshing completely eliminates the influence of prior-stage security parameters and integrates faithfully with a hardware-aware model of Barrett reduction. Using Lean 4, they complete 18 gap-free formal proofs and apply a combined structural-dependence and probing model to diagnose hardware accelerators. The methodology successfully identifies a first-order leakage in Microsoft’s Adams Bridge PQC accelerator caused by missing inter-stage refreshing, aligning with empirical observations.
📝 Abstract
This is Paper 6 of a series of formally-verified analyses of masked NTT hardware for post-quantum cryptography; Paper 1 [1] established structural dependency analysis of the QANARY platform, and Paper 2 [2] quantified security margins under partial NTT masking. Boolean masking composition is well-understood through NI, SNI, and PINI. Arithmetic masking over $\mathbb{Z}_q$ for prime $q$, the foundation of NTT-based post-quantum cryptography, has lacked an analogous theory. We prove, to our knowledge, the first machine-checked composition theorems for arithmetic masking over prime fields. Our key insight is the renewal argument: when a fresh random mask is applied between two pipeline stages, the intermediate wire becomes perfectly uniform regardless of Stage 1's security parameter. For two PF-PINI gadgets with parameters $k_1$ and $k_2$, the composed two-stage pipeline with fresh masking satisfies PF-PINI($k_2$), Stage 1's multiplicity is completely erased from the composed output. Without fresh masking, intermediate wires have multiplicity up to $k_1$, creating a necessary condition for differential power analysis. We formalize both theorems in Lean 4 with 18 machine-checked proofs and zero sorry stubs. We formally bridge the algebraic and hardware-faithful arithmetic models of Barrett reduction, and instantiate the theorems to formally diagnose Microsoft's Adams Bridge PQC accelerator: its absence of fresh inter-stage masking leaves Barrett output wires non-uniform under the first-order probing model, the same architectural flaw that two independent empirical analyses [3, 4] and our own prior structural analysis [1] identified. Computational evidence further suggests the 1-Bit Barrier is universal across Barrett and Montgomery reductions.