Abstract We describe the indecomposable ( v , 3)-matrices such that the systems of imprimitivity of the translation group of the space V v = {0, 1} v weakly diffused by the operation of modular addition remain invariant under multiplication by such matrices. It is shown that the use of such matrices has a substantial affect on the security of LAX-algorithms.
Abstract Let 𝔖 n be the semigroup of mappings of an n -element set X into itself and V n ( A ) be the set of mappings from the set 𝔖 n whose contours are of sizes belonging to the set A . Such objects are usually called A -mappings. It has been demonstrated earlier that the sequence | V n ( A )| n − n varies regularly as n tends to infinity if the set A has a positive asymptotic density in the set of positive integers. In this paper for a narrower class of sets A this assertion is refined with a power reduction in the degree of the remainder.
Abstract It is shown that for an arbitrary M -element subset of the Boolean n -cube there exists a linear hash function with clusters consisting of at most a elements and with rank not higher than 2 log 2 M − 2 log 2 a + 𝒪(1).
Abstract We find the limit joint distribution of the statistics T 1 , T 2 , T 3 of the «Monobit Test», «Frequency Test within a Block», and the generalized «Approximate Entropy Test» of the NIST package in the case where the tested sequence consists of independent Bernoulli random variables with parameter p = 1 2 . $p=\frac{1}{2} .$ We show that T 1 and ( T 2 , T 3 ) are asymptotically uncorrelated, T 2 and T 3 are asymptotically positively correlated, and T 1 , T 2 , T 3 are pairwise asymptotically dependent. We also prove that the covariance matrix C of the limit distribution of the vector ( T 1 , T 2 , T 3 ) satisfies C 12 = C 21 = C 13 = C 31 = 0, C 23 = C 32 > 0. The limit behavior of the vector ( T 1 , T 2 , T 3 ) is described in the case p ≠ 1 2 . $p \neq \frac{1}{2} .$
Abstract We consider the problem of synthesis of 3-pole contact circuits with poles A , B , and V implementing given Boolean functions between the poles A and B and admitting short fault detection tests with respect to contact breaks. For each n -place Boolean function and each test pole set containing at least one of the pairs { A , V }, { B , V }, we find the smallest possible lengths of the single and complete fault detection tests. In particular, it is proved that these lengths do not exceed 3.
Abstract Nonlinearity of a vectorial function is defined as the Hamming distance to the set of affine mappings. A connection between nonlinearity parameters and Fourier coefficients of the characters of a vectorial function is established. This connection is used to expose the possibility of finding nonlinearity parameters of a mapping by similar parameters of its components under various types of decomposition. A universal upper bound for nonlinearity is presented, expressions for the boundaries of nonlinearity are obtained in terms of Fourier coefficients of the characters, which make it possible to refine previously known boundaries for some classes of mappings. The dependence of the lower bound of nonlinearity on differential uniformity is found.
We obtain upper and lower estimates for the difference characteristics of permutations of the field F 2 n $\mathbb{F}_{2^{n}}$ whose restrictions to cosets of the group F 2 n × $\mathbb{F}^{\times}_{2^{n}}$ by its subgroup H , | H | = l , l · r = 2 n − 1, are mappings of the form x → c i x , c i ∈ F 2 n × , i = 0 , … , r − 1. $x\to c_{i}x,\, c_{i}\in\mathbb{F}^{\times}_{2^{n}},\, i=0,\dots,r-1.$
We prove that the complexity of computation of the threshold symmetric function T n n − 1 $T_n^{n-1}$ by monotone switching networks is Ω ( n log log n ).
We characterize the implicatively implicit extensions of all 27 two-place symmetric functions of the three-valued logic that preserve all three constants. We show that they contain extensions coinciding with the known implicatively closed classes and extensions which are not closed relative to the superposition operation.
Only finite groups are considered. A class of groups is called a formation if it is closed under taking homomorphic images and subdirect products. For a nonempty class Omega of simple groups, V. A. Vedernikov defined Omega-foliated formations of finite groups using two types of functions, viz., satellite functions and direction functions. Let sigma Omega be an arbitrary partition of the class Omega. We study sigma Omega-foliated formations, where sigma Omega is an arbitrary partition of the class Omega constructed by the authors of the preset paper as a natural generalization of the concept of an Omega-foliated formation using A. N. Skiba's sigma-methods. We prove the existence of different types of satellites of sigma Omega-foliated formations and describe their structure.
We find the limit joint distribution of the statistics T-1, T-2, T-3 of the "Monobit Test", "Frequency Test within a Block", and the generalized "Serial Test" tests of the NIST package in the case where the sequence under consideration consists of independent Bernoulli random variables with parameter p=1/2. We show that T-1 and (T-2, T-3) are asymptotically uncorrelated, T-2 and T-3 are asymptotically positively correlated, and T-1, T-2, T-3 are pairwise asymptotically dependent. We also prove that the covariance matrix C of the limit distribution of the vector (T1, T2, T3) satisfies C-12 = C-21 = C-13 = C-31 = 0, C-23 = C-32 > 0. The limit behavior of the vector (T-1, T-2, T-3) is described in the case p not equal 1/2.
We propose an algorithm for string processing which is based on finite ternary quasigroups. Ternary quasigroups are specified by three-dimensional matrices of a certain form. We prove sufficient conditions for simplicity and polynomial completeness of a finite ternary quasigroup. We also propose a method for deciding polynomial completeness of a finite ternary quasigroup based on the analysis of the corresponding three-dimensional matrix.
We study (v, k)-configurations with k = 5, which are combinatorial objects. We prove necessary and sufficient conditions for combinatorial equivalence of (v, 5)-configurations constructed from digraphs with two input and two output arcs in each vertex. We develop an algorithms for construction of (v, 5)-configurations and identification of combinatorially equivalent of (v, 5)-configurations. We also give a description of all (v, 5)-configurations for v <= 10 and evaluate the number of combinatorially nonequivalent (11, 5)-configurations.
For Boolean functions f of special form, we obtain an upper estimate for the length D(f) of a fault detection test if f is implemented by circuits of gates in the Zhegalkin basis with constant type-1 faults at gate outputs. As a corollary, D(f)<= nk-1(k-2)!+1 $D(f) \leq \frac{n<^>{k-1}}{(k-2)!}+1$ for functions f of n >= k variables whose Zhegalkin polynomial is of degree at most k.
We prove that the complexity of computation of the threshold symmetric function T-n(n-1) by monotone switching networks is Omega(n log log n).
It is proved that the generalized Wiener attack on the RSA cryptosystem allows one to find not only small, but also some large secret exponents d, and the fraction of exponents d which are weak against this attack is heuristically estimated as O(N-1/2).
We obtain upper and lower estimates for the difference characteristics of permutations of the field F-2n whose restrictions to cosets of the group F-2n(x) by its subgroup H, |H| = l, l . r = 2(n) - 1, are mappings of the form x -> c(i)x, c(i) is an element of F-2n(x), i=0,...,r-1.
We prove that a solution of functional equations of generalized transitivity for strongly dependent binary operations can be described similarly to the case of quasigroups by replacing the emerging group structure with a monoid. We also perform a generalization to the case of n-ary strongly dependent operations.
Aho-Corasick automaton is widely used to find occurrences of words from a given set in a text. In our paper we introduce an equivalence relation similar to R $\stackrel{R}{\sim}$ on states of Aho-Corasick automaton and prove indistinguishability of similar to R $\stackrel{R}{\sim}$-equivalent states. We also propose an algorithm for construction of a similar to R $\stackrel{R}{\sim}$-minimal automaton whose states are similar to R $\stackrel{R}{\sim}$-equivalence classes. Time and space complexity of this algorithm are linear in the number of states of the original Aho-Corasick automaton. Finally we consider cases in which the relations of similar to R $\stackrel{R}{\sim}$-equivalence and indistinguishability are identical, and thus the proposed automaton is minimal.
In the case when the tested sequence consists of independent random variables having a Bernoulli distribution with the parameter p=12 $p=\frac{1}{2}$ the limit joint distribution of the statistics T1, T2, T3 of the following three tests of the NIST package is obtained: << Monobit Test >>, << Frequency Test within a Block >> and << Binary Matrix Rank Test >>. Necessary and sufficient conditions for asymptotic uncorrelatedness and/or asymptotic independence of these statistics are obtained. It is proved that the covariance matrix C = & Vert;Cij & Vert; of the limit distribution of the vector (T1, T2, T3) satisfies the relations C12 = C21 = C13 = C31 = 0, C23 = C32 >= 0. The limit behavior of the vector (T1, T2, T3) is described for a wide class of values p not equal 12. $p\neq\frac{1}{2} .$
