Paper 2026/2236

A Complete Classification of Whole-Output Linear Structures in FEILIAN-Type Components

Abstract

We determine all input differences with constant whole-output XOR derivatives for a four-word ARX topology containing FEILIAN SubColumn. For every word width $w\geq2$ and arbitrary XOR-linear internal maps, the complete linear-structure space contains a universal two-dimensional MSB family and has dimension at most five, with equality attained. We give a necessary-and-sufficient classifier involving at most 32 candidate masks and characterize all additional structures, including an exceptional coset possible only when the first map's image lies in the span of the least and most significant bits. For the specified 64-bit FEILIAN component, the complete space is exactly the universal family. With FEILIAN's actual ShiftRow, no nonzero member remains constant after both of the first two SubColumn layers. We also classify propagation at every boundary for all 20,160 invertible binary four-row mixers composed with ShiftRow. The five profiles stabilize by the third SubColumn layer, and exactly 4,032 mixers (20%) retain a nonzero family indefinitely. Algebraic proofs are supplemented by exhaustive small-word checks and a separate census audit. These results concern component structures and constancy at successive boundaries; they neither classify all endpoint structures of the iterated permutation nor establish an attack on the hash.