In the work we consider the new algorithm of lowering pairs finding for functions without prohibition of four variables. Lowering (or restrictive) pairs of natural numbers (h,t), h>t for functions without prohibition was studied earlier in connection with the construction of bijective mappings Bf,L defined by the shift register of length n with feedback function L, essentially dependent on a limited number of initial and final arguments, and a nonlinear function of removal of k arguments (k<<n).
All investigations of the scheme defined in its name are fulfiled by the direct enumeration of its outcomes, namely: the number of its outcomes of the scheme and its probability distribution are defined, the numbering problem for outcomes of the scheme is solved, this gives the possibility of a quick modeling of its possible values and approximate calculation of the number of outcomes of the scheme by the method of proportion.
We decide the problems of finding of quantities of outcomes in the scheme of the arrangement of distinguishable and indistinguishable particles to distinguishable and indistinguishable cells under different restrictions on the levels of the filling of cells and we establish the order of enumeration admissible outcomes establish and we calculate their probability distribution.
The number of outcomes is found in the scheme under study by the method of an additional graph, all its outcomes are enumerated, their probability distribution is determined, the numbering problem for the outcomes of the scheme in direct and inverse formulations is solved, based on the results of which their quick modeling is proposed. A similar scheme without restrictions is used as a general scheme, and as an additional scheme, the same scheme is used with a restriction: each cell has at least a certain number of particles.
The generalization of thescheme of sequential actions belongs to the case of the dependence of the number of outcomes of each next action not only of the action but on the result of the previos action. We fulfiled the enumeration of outcomes of the scheme, the enume-ration problem is solved theoreticaly and two methods of modeling are suggested.
We study the questions of finding the number of permutations with a given information of structures of cycles, the direct enumeration and modelling of their outcomes.
The fulfilled analyses of scheme consistsof the establishment of the connection between the schemes with outcomes in the common traektorial from, in finding of the number of outcomes, in the con- straction of the procedur of their enumeration, in the solution of the enumeration problem and of the modeling of their outcomes.
We solve the problem of enumeration of all outcomes and enumeration, problem on classical scheme of permutations with repetition and close scheme with extend outcomes.
We consider different procedures to number all outcomes of a arrangement scheme, establish a one-to-one correspondence between the outcomes and its numbers generated in the numbering рrocedure, and give some methods to simulate the outcomes.
We consider procedures to numbers all outcomes of a combination scheme, establish a one-to-one correspondence between the outcomes and its numbers generated in numbering procedure, and give some methods to simulate the outcomes.
The direct enumeration of the outcomes by the graph method fulfils and their number is find, the probability of outcomes of the scheme in the general combination scheme. The enumeration is solved for outcomes of the scheme and on this base the algorithm of their quick modelling is suggested.
The notion of the range R of outcomes of a combination scheme is Introduced as the maximal difference between the chosen elements. The number of outcomes of the scheme with limited range R not grater t is defined, their enumeration isfulfilled and the problem of numeration (theproblem of finding one-to-one correspondence between the number of outcomes and their forms is determined) solved for them, modeling of outcomes of scheme is discussed.
Under the conditions of the known results for the more general combinatorial scheme with respect to the given scheme and the scheme of the rest its outcomes (the additional scheme) in the directios of the solution of the problems of the enumerational combinatorial analysis we suggest to obtain all results of its analysis from the corresponding results of the general and additional scheme with the consideration of the reject of the outcomes of the additional scheme in the graph of the enumeration of the outcomes of the general scheme instead of its direct investigation. Such approach further will be called the method of additional graph ( MAG).
In this paper we propose a new algorithm for verifying statistical indistinguishability (regarding probabilities of output s-grams) of sequences generated by functions f:Xn®F2 in the law
In this initial sequence x1,x2,… is viewed as a sequence of independent identically distributed random variables defined on the input alphabet X and takes values xÎX with probability px>0, Spx=1. This algorithm identifies the indistinguishability of the output for a given pair f,j functions of 2n variables, which are represented by the sum of two functions of n disjoint variables
The proposed algorithm uses modular operations on integers of standard length and has a much lower computational complexity compared to the previously known algorithm applied to the functions of the general form of 2n variables. The results may be useful in the construction and justification of the statistical properties of random sequences using the appropriate complicating conversion of intermediate scales.
The paper deals with the construction of orthogonal systems of Boolean functions (f(x),f(d(x)),...,f(dn-1(x)), xÎ(F2)n generated by conversion d=dL, carried out shift register of great length n with the function of feedback L and a nonlinear function f removal from a small number of arguments k (k<<n). The orthogonality of the specified system functions is equivalent to the mapping Вf,L, defined by a set of coordinate functions (f(x),f(d(x)),...,f(dn-1(x)), is bijective. The new method is proposed, which reduces the original problem to the verification of orthogonal systems of Boolean functions with shift registers limited length n<n0, which allows efficient use of its computing solutions. This method, in particular, allowed to build new infinite classes of bijective mappings Вf,L for the case of nonlinear function f, depending on four variables f=f(x1,x2,x3,x4). Earlier, similar results were known for the case when the function f depends on three arguments f=f(x1,x2,x3). The results can be useful for construction and proof of the statistical properties of the generating random sequences on the basis of filter generators. The particular practical importance has the choice of pairs (f,L), in which simultaneously the mapping Bf,L is bijective and period of dL is maximal.