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

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

On the Boomerang Uniformity of some Permutation Polynomials

The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt\u2718, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT) that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Chunming and Maosheng. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions