On the diffusion of the Improved Generalized Feistel

Abstract

<p style='text-indent:20px;'>We consider the Improved Generalized Feistel Structure (IGFS) suggested by Suzaki and Minematsu (LNCS, 2010). It is a generalization of the classical Feistel cipher. The message is divided into <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula> subblocks, a Feistel transformation is applied to each pair of successive subblocks, and then a permutation of the subblocks follows. This permutation affects the diffusion property of the cipher. IGFS with relatively big <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula> and good diffusion are of particular interest for light weight applications.</p><p style='text-indent:20px;'>Suzaki and Minematsu (LNCS, 2010) study the case when one and the same permutation is applied at each round, while we consider IGFS with possibly different permutations at the different rounds. In this case we present permutation sequences yielding IGFS with the best known by now diffusion for all even <inline-formula><tex-math id="M3">\begin{document}$ k\le 2048 $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M4">\begin{document}$ k\le 16 $\end{document}</tex-math></inline-formula> they are found by a computer-aided search, while for <inline-formula><tex-math id="M5">\begin{document}$ 18\le k\le 2048 $\end{document}</tex-math></inline-formula> we first consider several recursive constructions of a permutation sequence for <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula> subblocks from two permutation sequences for <inline-formula><tex-math id="M7">\begin{document}$ k_a< k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ k_b< k $\end{document}</tex-math></inline-formula> subblocks respectively. Using computer, we apply these constructions to obtain permutation sequences with good diffusion for each even <inline-formula><tex-math id="M9">\begin{document}$ k\le 2048 $\end{document}</tex-math></inline-formula>. Finally we obtain infinite families of permutation sequences for <inline-formula><tex-math id="M10">\begin{document}$ k>2048 $\end{document}</tex-math></inline-formula>.</p>