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Explicit permutations of $(\mathbb{Z}/4\mathbb{Z})^m$ with uniformly bounded differential uniformity — arXiv:2611.02572

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.02572
License (as archived): CC0-1.0
Archived: 2026-11-03T18:33:18.852Z
SHA-256 of the archived text: 4514b2a2760b17e236002930577c5bc117a2ca9aaaedb17eb55f4c4a766f9757
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How to cite

Explicit permutations of $(\mathbb{Z}/4\mathbb{Z})^m$ with uniformly bounded differential uniformity — arXiv:2611.02572 — arXiv (en), https://arxiv.org/abs/2611.02572 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bf80ddf809f7 (Metraipolis archive dc_bf80ddf809f7, sha256 4514b2a2760b17e236002930577c5bc117a2ca9aaaedb17eb55f4c4a766f9757)

BibTeX

@misc{dcbf80ddf809f7,
  title = {Explicit permutations of $(mathbbZ/4mathbbZ)^m$ with uniformly bounded differential uniformity — arXiv:2611.02572},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.02572},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bf80ddf809f7; sha256 4514b2a2760b17e236002930577c5bc117a2ca9aaaedb17eb55f4c4a766f9757},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.02572. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Symmetric 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 $N=q^2$. For every $m\ge1$ we construct an explicit permutation $G_c$ 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 $2/\sqrt{N}$ (Ronjom--Sandrib). The bound holds for the fixed permutation itself, not on average over keys. Equivalently, $\Delta(G_c)\le36$ against $\Delta\ge2\sqrt{N}$ for permutation polynomials; since every permutation has $\Delta\ge2$, the least differential uniformity of a permutation of $(\mathbb{Z}/4\mathbb{Z})^m$ lies between $2$ and $36$ for every $m$. In Teichmuller digits $(a,b)\in\mathbb{F}_q^2$ and for $c\in\mathbb{F}_q$ of absolute trace $1$, $G_c(a,b)=(L,(L^2+L+c)a)$ with $L=(a^2+a+c)b$. The map is $F_c\circ J\circ F_c$, where $F_c(a,b)=(a,(a^2+a+c)b)$ 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 $F\circ J\circ F$ with $F(a,b)=(p(a),h(a)+d(a)b)$, where $p,d,h$ have degrees bounded independently of $m$, $p'\ne0$ 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 ($\Delta\le440$, resp. $510$, for odd $m$).

Tema degli esploratori: fiducia digitale e crittografia (elenco arXiv cs.CR, nuovi lavori del giorno).
Categorie arXiv: Number Theory (math.NT) ; Cryptography and Security (cs.CR); Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2611.02572 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T18:12:23.826Z dalla pagina https://arxiv.org/list/cs.CR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).