Volume 35, Issue 1

Let p be a prime number, p ≥ 3. We consider the set of decompositions of a p -logic function into a product of functions with disjoint subsets of variables obtained by means of linear substitutions of arguments. Each decomposition of this kind is associated with a decomposition of the vector space into a direct sum of subspaces. We present conditions under which such space decomposition is unique up to rearrangement of subspaces. Also, a criterion for such product to be balanced is given.

We propose a new technique for dimensional analysis of the Hadamard (Schur) square of an error-correcting linear code. This is usually achieved by a representation of the Hadamard square as an image of some linear operator defined on the set of quadratic forms. A link between the dimension of the Hadamard square the rank of some submatrix of the generating matrix of the code containing the set of vector values of quadratic forms is established. So the dimensional analysis of the Hadamard square can be carried out with the extensive code-based machinery rather than via the approach with estimation of the number of joint zeros of the set of quadratic forms. As a result and we establish a nonasymptotic estimate for the probability that the Hadamard square of a random linear code fills the entire space. This estimate can be used for cryptographic analysis of post-quantum code-based cryptosystems.

We propose a method to construct permutations of the field F2n $\mathbb{F}_{2^{n}}$ with low value of the Δ -uniformity such that their restrictions to the cosets of the group F2n× $\mathbb{F}^{\times}_{2^{n}}$ by some of its subgroup H are linear. Using the proposed method, we construct a large number of new CCZ-nonequivalent differentially 4-uniform permutations, orthomorphisms, and involutions over the field F2n $\mathbb{F}_{2^{n}}$ with n = 6, 8.