Volume 33, Issue 2

Nonlinear invariants of round transformations in XSL-schemes are studied. The emphasis is on invariants which may be found by means of the approach suggested at the conference ASIACRYPT 2016. Some known results on the inertia groups of decomposable functions are used to describe conditions on S-boxes and matrices of XSL-schemes which are necessary for the existence of such invariants. It is shown that for a number of schemes these conditions are not satisfied.

We consider the branching process Zn=Xn,1+⋯+XnZn−1 $ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $, in random environments η , where η is a sequence of independent identically distributedvariables, for fixed η the random variables X i , j areindependent, have the geometric distribution. We suppose that the associated random walk Sn=ξ1+⋯+ξn $ S_n = \xi_1 + \dotsb + \xi_n $ has positive mean μ ,0 <  h < h + satisfies the right-hand Cramer’s condition E exp( h ξ i ) < ∞ for, some h + . Under theseassumptions, we find the asymptotic representation for local probabilities P ( Z n =⌊exp( θ n )⌋) for θ  ∈ [ θ 1 , θ 2 ]⊂</ given − names >< x > </ x >< surname >( μ ; μ + ) and some μ + .

V. I. Solodovnikov had employed the shift-composition operation to investigate homomorphisms of shift registers into linear automata; in his papers, conditions for the absence of nontrivial inner homomorphisms of shift registers were derived. An essential role was played by the condition of linearity of the left component of the shift-composition operation in the corresponding polynomial decomposition. In this paper we consider the case where the left component is a function belonging to a wider class, which includes the class of linear functions.

In this paper, we give a method to determine a complete set of mutually orthogonal Latin squares of order m , where m is an odd prime or power of a prime, as a group transversal of a Frobenius group.

We construct an explicit finite basis for admissible rules in K -saturated tabular logics that extend the logic Grz .

An n -place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n , the maximally nonlinear functions are bent functions, but, for q  > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.