cronokirby

(2026-08) Five-Bit Axial Subspace Differential Uniformity; Exact Optima and a DDT-Support Realizability Gap

2026-08-12

Abstract

Subspace differential uniformity (SDU) measures the concentration of a differential distribution table (DDT) on affine subspaces. We determine the exact optima of the two axial affine-SDU coordinates in dimension five and establish a strict gap between locally admissible support designs and supports realizable by almost perfect nonlinear (APN) permutations. We first prove that every 1616-subset of F25\mathbb{F}_2^5 meets some affine 33-flat in at least six points. Equality holds precisely for balanced quadratic indicators of polar rank four, forming a single affine orbit represented by supp⁡(Tr⁡(x3))\operatorname{supp}(\operatorname{Tr}(x^3)). This gives the relaxed axial optimum 1212. We then classify all 900900 ordered monomial-trace candidates Mr,s(u,v)=Tr⁡(urvs)M_{r,s}(u,v)=\operatorname{Tr}(u^r v^s): exactly 100100 attain both relaxed axial optima, yielding 2020 labelled arrays and three product-linear types up to transpose. Despite satisfying regularity and zero-vector-sum constraints, every optimal type violates a global necessary condition for DDT realizability: its two-dimensional character transform contains a negative coefficient where a vectorial Walsh square is required. Finally, the published exhaustive affine classification of five-bit APN permutations, together with direct recomputation of all five class representatives, gives realizable axial minimum 1414, attained simultaneously by the x15x^{15} class. Thus local incidence and vector-sum constraints permit score 1212, whereas genuine APN-permutation DDT supports require score 1414.