🤖 AI Summary
This study investigates whether stage-wise arithmetic masking in number-theoretic transform (NTT)-based post-quantum cryptographic hardware pipelines ensures end-to-end side-channel security under r-bearing scenarios with fresh randomness. For the first time, we formally verify in Lean 4 and Mathlib the arithmetic masking scheme of Cooley–Tukey NTT butterfly operations over ℤ_q, proving that a k-stage pipeline satisfies stage-wise uniformity under the ISW first-order probing model, yet fails to guarantee pointwise value independence. Our work bridges the formal verification gap for compositional security of arithmetic masking beyond Boolean domains and exposes a structural vulnerability in the Adams Bridge accelerator stemming from its violation of the fresh-masking assumption. All nine theorems are fully machine-verified with zero “sorry” placeholders across 1,738 build tasks.
📝 Abstract
Post-quantum cryptographic (PQC) accelerators for ML-KEM (FIPS 203) and ML-DSA (FIPS 204) rely on pipelined Number Theoretic Transform (NTT) stages over $\mathbb{Z}_q$. Our prior work established structural dependency analysis at scale [1] and quantified the security margin of partial NTT masking [2]. Whether per-stage arithmetic masking guarantees pipeline-level security had no prior machine-checked answer for the r-bearing case: composition frameworks (ISW, t-SNI, PINI, DOM) were formalized exclusively for Boolean masking over $\mathrm{GF}(2)$; no proof assistant artifact addresses the NTT butterfly over $\mathbb{Z}_q$. We present three machine-checked results in Lean 4 with Mathlib, all zero sorry. First, we close a stated limitation of prior work: value-independence implies constant marginal distribution under fresh randomness (via an algebraic MutualInfoZero proxy). Second, butterfly per-context uniformity: for any Cooley-Tukey butterfly with fresh output mask over $\mathbb{Z}/q\mathbb{Z}$ ($q > 0$), each output wire has exactly one mask value producing each output, a uniform marginal independent of secrets, universal over all moduli, twiddle factors, and inputs. Third, a k-stage NTT pipeline with fresh per-stage masking satisfies per-context uniformity at every stage under the ISW first-order probing model. We document a named warning: pointwise value-independence is false for butterfly outputs. The Adams Bridge accelerator (CHIPS Alliance Caliptra) fails the fresh masking hypothesis, masking active only in INTT round 0, architecturally explaining its structural insecurity. Artifact: nine theorems, 1,738 build jobs, zero sorry. Composition for nonlinear gadgets (Barrett) is addressed in forthcoming manuscripts proving Barrett's PF-PINI(2) satisfaction ('one-bit barrier') [3] and k-stage composition for PF-PINI gadgets under fresh-mask renewal [4].