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The 1-Bit Barrier is Universal: k-Stage Pipeline Composition and Unified Leakage Bounds for Masked NTT Hardware
Arbitrary-depth k-stage masked NTT pipelines with fresh inter-stage masking and per-stage PF-PINI(at most 2) gadgets satisfy a per-observation cardinality bound of 2 times q^(2k-2) on the preimage of any output value, machine-checked in Lean 4 with zero sorry. Under the standard (informal) semantic translation that divides this cardinality by the total mask-tuple space size q^(2k-1), the per-observation conditional probability bound is 2/q, independent of pipeline depth k. This paper closes the QANARY program with four contributions. First, a k-stage composition theorem generalizing the earlier two-stage result to arbitrary k, giving a last-stage-determined bound in which only the last stage's PF-PINI parameter survives, with intermediate parameters erased by fresh inter-stage masking. Second, Montgomery reduction is shown to satisfy PF-PINI(2), matching Barrett. Third, a single end-to-end cardinality bound assembling the program's gadget results. Fourth, a discussion of the cardinality-versus-probability semantic gap, the first-order probing scope, and the absence of a formal translation between PF-PINI and PINI.
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@misc{iskander2026onebit,
title = {The 1-Bit Barrier is Universal: k-Stage Pipeline Composition and Unified Leakage Bounds for Masked NTT Hardware},
author = {Ray Iskander and Khaled Kirah},
year = {2026},
}
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