Explicit permutations of (Z/4Z)^m with uniformly bounded differential uniformity

Symmetric primitives over Galois rings need nonlinear components whose differentials with respect to ring addition are controlled. Let R = GR(4,m) and q = 2m, so that (R,+) ≅ (Z/4Z)m has order N = q2. For every m ≥ 1 we construct an explicit permutation Gc of R all of whose differentials with a nonzero input difference have probability at most 36/N, whereas every permutation polynomial of R has such a differential of probability at least 2/√N (Rønjom–Sandrib). The bound holds for the fixed permutation itself, not on average over keys. Equivalently, Δ(Gc) ≤ 36 against Δ ≥ 2√N for permutation polynomials, so the least differential uniformity of a permutation of (Z/4Z)m lies between 2 and 36 for every m. In Teichmüller digits (a,b) ∈ Fq2 and for c ∈ Fq of absolute trace 1, Gc(a,b) = (L, (L2+L+c)a) with L = (a2+a+c)b. The map is Fc∘J∘Fc, where Fc(a,b) = (a,(a2+a+c)b) is induced by a polynomial over R and J exchanges the digits: a single digit exchange between two polynomial layers yields a bound independent of m. The proof reduces the difference equations, in each of three types of directions, to one-variable polynomials whose degrees do not depend on m. For general maps F∘J∘F with F(a,b) = (p(a), h(a)+d(a)b), where p, d, h have degrees bounded independently of m, p′ ≠ 0 and d has no zeros, a Jacobian identity in characteristic 2 bounds the differentials with nonzero low input and output differences independently of m; the remaining directions need a separate argument, which we give for an explicit family (Δ ≤ 440, resp. 510, for odd m). Files. manuscript.pdf is the paper (8 pages). z4m-bounded-differential-uniformity-1.0.0.zip is the archive of release v1.0.0 of https://github.com/hirokifukui/z4m-bounded-differential-uniformity: the TeX source, the Magma scripts and logs of Appendix A, the C program and fail-closed runner for Table 1 with the 11 recorded runs, the manifest with SHA-256 hashes, and REPRODUCE.md. The proofs do not depend on these computations. Licences. Text (paper, README, REPRODUCE, CITATION.cff): CC BY 4.0. Scripts, programs and recorded outputs: MIT. See LICENSE in the archive.