Permutation of Z/4Z^m with Bounded Differential Uniformity
Original title: Explicit permutations of $(\mathbb{Z}/4\mathbb{Z})^m$ with uniformly bounded differential uniformity
Explicit G_c permutation over Galois rings achieves uniform differential uniformity bound of 36 independent of m.
Symmetric primitives require controlled differentials for nonlinear components over GR(4,m). The constructed G_c permutation ensures all nonzero input differences have differential probability at most 36/N. This improves on prior permutation polynomials that reach at least 2/sqrt(N). The map is defined via Teichmuller digits and a single digit exchange between polynomial layers.
- Bound holds for fixed permutation not average over keys
- Delta(G_c) <= 36 against Delta >= 2 sqrt(N)
- Least differential uniformity lies between 2 and 36
- Construction via F_c o J o F_c
- Proof reduces to one-variable polynomials independent of m
From the original article
1 min read at arxiv.orgSymmetric primitives over Galois rings need nonlinear components whose differentials with respect to ring addition are controlled. Let $R=\mathrm{GR}(4,m)$ and $q=2^m$, so that $(R,+)\cong(\mathbb{Z}/4\mathbb{Z})^m$ has order $N=q^2$.
By Hiroki Fukui · arxiv.org
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