О способе построения дифференциально 2δ-равномерных подстановок на F22m

Рассмотрены способы построения дифференциально 25-равномерных подстановок на F22m для случая m ^ 3. Предложенный подход излагается с использованием так называемого TU-представления функций и обобщает известный способ построения дифференциально 4-равномерных подстановок поля F22 m с применением подстановки обращения ненулевых элементов поля. F22m. The paper studies new ways of constructing differentially 24-uniform bijections over F22m, m ^ 3, that are based on TU-con- struction. Some well known results on the constructing differentially 4-uniform permutations over F22m are generalized in this work. The core idea is to use TU-construction and differentially 4-uniform bijections to construct 2* • 4-uniform permutations. A generalized method for constructing 2m-bit differentially 4-uniform permutations is proposed, and new constructions of differentialy 6 and 8-uniform permutations are introduced

Private Graphon Estimation for Sparse Graphs

We design algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members. Given a sparse input graph $G$, our algorithms output a node-differentially-private nonparametric block model approximation. By node-differentially-private, we mean that our output hides the insertion or removal of a vertex and all its adjacent edges. If $G$ is an instance of the network obtained from a generative nonparametric model defined in terms of a graphon $W$, our model guarantees consistency, in the sense that as the number of vertices tends to infinity, the output of our algorithm converges to $W$ in an appropriate version of the $L_2$ norm. In particular, this means we can estimate the sizes of all multi-way cuts in $G$. Our results hold as long as $W$ is bounded, the average degree of $G$ grows at least like the log of the number of vertices, and the number of blocks goes to infinity at an appropriate rate. We give explicit error bounds in terms of the parameters of the model; in several settings, our bounds improve on or match known nonprivate results.Comment: 36 page

Differentially 4-Uniform Bijections by Permuting the Inverse Function

Block ciphers use Substitution boxes (S-boxes) to create confusion into the cryptosystems. Functions used as S-boxes should have low differential uniformity, high nonlinearity and algebraic degree larger than 3 (preferably strictly larger). They should be fastly computable; from this viewpoint, it is better when they are in even number of variables. In addition, the functions should be bijections in a Substitution-Permutation Network. Almost perfect nonlinear (APN) functions have the lowest differential uniformity 2 and the existence of APN bijections over \F_{2^n} for even $n\ge 8$ is a big open problem. In the present paper, we focus on constructing differentially 4-uniform bijections suitable for designing S-boxes for block ciphers. Based on the idea of permuting the inverse function, we design a construction providing a large number of differentially 4-uniform bijections with maximum algebraic degree and high nonlinearity. For every even $n\ge 12$, we mathematically prove that the functions in a subclass of the constructed class are CCZ-inequivalent to known differentially 4-uniform power functions and to quadratic functions. This is the first mathematical proof that an infinite class of differentially 4-uniform bijections is CCZ-inequivalent to known differentially 4-uniform power functions and to quadratic functions. We also get a general lower bound on the nonlinearity of our functions, which can be very high in some cases, and obtain three improved lower bounds on the nonlinearity for three special subcases of functions which are extremely large

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