Differentially low uniform permutations from known 4-uniform functions

Functions with low differential uniformity can be used in a block cipher as S-boxes since they have good resistance to differential attacks. In this paper we consider piecewise constructions for permutations with low differential uniformity. In particular, we give two constructions of differentially 6-uniform functions, modifying the Gold function and the Bracken–Leander function on a subfield.publishedVersio

A New Method to Investigate the CCZ-Equivalence between Functions with Low Differential Uniformity

Recently, many new classes of differentially $4$-uniform permutations have been constructed. However, it is difficult to decide whether they are CCZ-inequivalent or not. In this paper, we propose a new notion called Projected Differential Spectrum . By considering the properties of the projected differential spectrum, we find several relations that should be satisfied by CCZ-equivalent functions. Based on these results, we mathematically prove that any differentially $4$-uniform permutation constructed in \cite{CTTL} by {C.Carlet, D.Tang, X.Tang, et al.,} is CCZ-inequivalent to the inverse function. We also get two interesting results with the help of computer experiments. The first one is a proof that any permutation constructed in \cite{CTTL} is CCZ-inequivalent to a function which is the summation of the inverse function and any Boolean function on \gf_{2^{2k}} when $4\le k\le 7$. The second one is a differentially $4$-uniform permutation on \gf_{2^6} which is CCZ-inequivalent to any function in the aforementioned two classes

Permutations via linear translators

International audienceWe show that many infinite classes of permutations over finite fields can be constructedvia translators with a large choice of parameters. We first characterize some functionshaving linear translators, based on which several families of permutations are then derived. Extending the results of \cite{kyu}, we give in several cases thecompositional inverse of these permutations. The connection with complete permutations is also utilized to provide further infinite classes of permutations. Moreover, wepropose new tools to study permutations of the form $x\mapsto x+(x^{p^m}-x+\delta)^s$ and a few infinite classes of permutations of this form are proposed

A generalisation of Dillon's APN permutation with the best known differential and linear properties for all fields of size 24k+22^{4k+2}

The existence of Almost Perfect Nonlinear (APN) permutations operating on an even number of variables was a long-standing open problem, until an example with six variables was exhibited by Dillon et al. in 2009. However it is still unknown whether this example can be generalised to any even number of inputs. In a recent work, Perrin et al. described an infinite family of permutations, named butterflies, operating on (4k+2) variables and with differential uniformity at most 4, which contains the Dillon APN permutation. In this paper, we generalise this family, and we completely solve the two open problems raised by Perrin et al.. Indeed we prove that all functions in this larger family have the best known non-linearity. We also show that this family does not contain any APN permutation besides the Dillon permutation, implying that all other functions have differential uniformity exactly four

Sparse Permutations with Low Differential Uniformity

National audienc