Introduction

It is well known, that the main advantage of non-positional notation in the residue classes (NRC) is laid in the possibility of quick process of organization of modular arithmetic operations of addition, subtraction and multiplication [1,2].

However, in computer systems for common purpose it is needed to perform socalled non-modular (position) operations except the above listed arithmetic operations. The following operations belong to these: ,..., , , ,..., ,

i i i n n А а а а а a a   

. It is appropriate to use so-called positional indications of non-positional code (PINC) to define an index of j numerical interval of detecting A. The following indications are the most frequently used in NRC (within existing PINCs) [3,4]:

 Indications based on the procedure of number's translating from NRC to PINC;  Indications based on the procedure of nulevization (reduction to zero polynomial) (definition of the value of 1 n y  );  Indications based on the procedure of expanding given system of basis of NRC;  Rank r of number A, etc.

There are disadvantages of the above-mentioned PINC. At first, it is technical and time complexity of PINC generating (developing) for given code structure

  1 -1 1 1

,..., , , ,..., ,

i i i n n А а а а а a a    .

At second, it is no mean time of implementation having applied the existing positional indications, non-modular operations in NRC, specifically, operations of control, diagnostic of correction of data mistakes [3,4]. It becomes obvious that the research of developing methods of generating new PINC in NRC is important; because of them, we can use operatively non-modular operations. It should be pointed out, population (sequence) of defined modular and non-modular operations, which are implemented by PINC, can realize any nonmodular operation.

Beforehand we will consider the general requirements to PINC, on base of which in article will be developed method of increasing of operational efficiency of data control:

 To define exactly correctness and incorrectness of number A in NRC (to define fact of detecting/not detecting of number A in an informational numerical [0, ] M interval, where

1 n i i M m   

) by used (chosen, developed, generated) PINC;

 Simplicity of generating PINC for given code structure of data

  1 -11 1

,..., , , ,..., ,

i i i n n А а а а а a a  

 ;  Simplicity of using of generated indication for data control in NRC (in general case for implementing positional operations);  Indication has to have clear and understandable physical meaning;  Indication must be analytically described by simple mathematical relator;  It is possible to technically implement process of data control in NRC by using PINC;  Using the chosen indication of non-positional code has to provide the given fidelity of data control in NRC;

 PINC using has to provide the possibility of exception from the control procedure, diagnostic and correction of mistakes in NRC the most difficult positional operations.

It is reasonable to develop and research the method of data control in NRC on the base of PINC using, resulting from the above-mentioned information.

Method of generating of positional indication of nonpositional code structure of data in NRC

Looking upon method of generating of PINC (Fig. 1). The indication A n forms proceeding from representation of initial code structure

  1 -11 1

,..., , , ,..., ,

i i i n n А а а а а a a    , which is represented as NRC with ba- sis { } i m , 1, 1 i n 

 , in the way of creating of so-called uniserial code (UC).

In standard form, UC

  ( ) ( ) ( ) ( ) ( ) 1 1 1 0 ... A n A A A A N N N K Z Z Z Z   

represents sequence of binary bits, which contains ones and a zero, index of which is at A n place (the count is performed from right to left, from digit bit ( )

0 A Z to digit bit ( ) 1 ). A N Z  Indication of PINC A n defines an index of j numerical interval    , 1 i i jm j m    of detecting of number   1 -11 1

,..., , , ,..., ,

i i i n n А а а а а a a    .

Mathematically PINC A n represents a positive integer, which points out the location of zero binary bit in the record of UC  

( ) ( ) 0 A A n A N n K Z  .

The procedure of generating UC ( )

A n N K lies in the following. A constant   ( ) ' ' ' ' 1 1 1 1 ,..., , , ,..., i A i i i n m KH a a a a a    

defines in the block of constants of nuvelization (BCN). It is performed by the definition of remainder i a of number

  1 -11 1

,..., , , ,..., ,

i i i n n А а а а а a a    for chosen inte- ger m i of NRC.

Further, using the chosen constant of nuvelization

( ) i A m KH

, we displace number A on the left edge of the interval

   , 1 i i jm j m    by implementing operation     ' ' ' ' ' 1 2 1 1 1 1 2 1 1 1 (1) (1) (1) (1)(1i i i n i i i n i i n i i A m m A A KH a a a a a a a a a a a a a a a a a                    1. Choice informational { } i m , 1, i n  and control 1 k n m m   ( 1 i i m m  

) basis of residue classes for representing data

1 2 1 1 1

( , ,..., , , ,..., , )

i i i n n A a a a a a a a     , GCD ( , ) 1 i j m m  , i j  .

Choice of basis { }i i m m  , ( 1, 1) j n   , by which defines an index of j numerical interval    , 1 i i jm j m   

of detecting number А.

3.

Defining the constant of reducing to zero polynomial 4.

Defining the value ( )i i A m m A A KH      1 2 1 1 1

, ,..., , , ,..., ,

i i i n n a a a a a a a       (1) (1) (1) (1) (1) ' ' ' '(1) 1 2 1 1 1 1 2 1 1 1 , ,..., , , ,..., , , ,..., , 0, ,..., ,

.

i i i n n n i i n a a a a a a a a a a a a a             5. Defining UC   ( ) ( ) ( ) ( ) 1 2 0 ... A n A A A N N N K Z Z Z    ,   ( ) ( ) ( ) ( ) 0 1 2 ... A i i i n A A A N N N K Z Z Z    . 1 1 n K K K i N m      , ] / [ i i N M m  , 1 n i i M m    . ( ) i A A m A i K A K m Z    . ( ) 0 ( ) 1 ( ) 2 ( ) 1 0 , 1 , . . . ( 2) ,( 1)

.

i i i i A m i A m i A m i N A m i N A m Z A m Z A N m Z A N m Z                        ( ) 0 ( ) 1 ( ) 2 ( ) 1 0 , 1 , . . . ( 2) ,( 1)

. M . We have to implement operation

i i i i i i A m i A m i A m i i N A m i i N A m Z A m Z A N m Z A N m Z                        6 Defining PINC A n of number A, which means numerical value of A n for which ( ) ( ) 0 A A A A K n Z Z   , which means 0 i m A i A n m    . Herewith ( ) 1, ( 0; ) i A m i A l Z A l m l n      .( ) i A A m A i K A K m Z   ( 1 )

to define the fact of detecting number in numerical informational interval [0, M). The operation ( 1) is carried out simultaneously using the population of N constants

  0, 1 A i A K m K N    , where 1 1 n K K K i N m      : ( )0 ( ) 1 ( ) 2 ( ) 2 ( ) 1 0 , 1 , 2 , . . . ( 2) , ( 1)

.

i i i i i A m i A m i A m i A m i N A m i N A m Z A m Z A m Z A N m Z A N m Z                            ( 2 )

In this case, UC represents in the view

  ( ) ( ) ( ) ( ) ( ) 1 2 1 0 ... A n A A A A N N N K Z Z Z Z   ( 3 )

The only value A n from (1) exists in total (2) analytical rations. For A n there is ( ) ( )

( )A A A A A A K n Z Z K n    , which means 0 i m A i A n m    . The rest of values (2) equal ( ) 1 A l Z  ( 0 ; i m i A l m    ) A l n  .

In the general case, the amount of N binary bits in the record UC ( )

A n N K equal to value 1 1 n K K K i N m      .

It should be pointed out, that there is no necessity to have the whole sequence of values ( ) In this case values of variables of numerical intervals

A A K Z from(   , 1 i i jm j m   

, which are out of informational interval   0, M , make no matter for establishing the fact of correctness control of number A.

Method of operational data control in NRC on base of using positional indication of non-positional code

The procedure of generating of positional indication of non-positional code (fig. 1) laid in the base of method of operational data control in residue classes. In that way, the essence of the method of data control in residue classes lies in the following. For the controlled code structure

  1 -11 1

,..., , , ,..., ,

i i i n n А а а а а a a    , which is repre- sented in residue classes, developed (defined) PINC A n by generating UC   ( ) ( ) ( ) ( ) ( ) 1 0 1 2 ... A i i i n A A A A N N N K Z Z Z Z    in view    ( ) ' ' ' ' 1 2 1 1 2 1 (1) (1) (1) (1) 1 2 1

, ,..., ,..., , , ,..., ,..., , , ,..., 0,..., , .

i i A m i n n i n n m n n A A KH a a a a a a a a a a a a a a             Using Ni constants A i K m    0, 1 A i K N   simultaneously the operations of sub- tractions i m A i A K m  

are carried out, in the result of which appears the values of binary bits ( )

A A K Z , so the UC ( ) A i n N K forms.

The values of PINC A n defined from the equation 0

i m A i A n m    . If A i n N 

, number A is considered as a wrong. In the opposite case

( A i n N 

) number A is correct. In common view, method of data control is represented on Fig. 2.

Algorithm of defining UC   ( ) ( ) ( ) ( ) ( ) 1 0 1 2 ... A i i i n A A A A N N N K Z Z Z Z    of number   1 2 1 1 1

, ,..., , , ,..., ,

i i i n n A a a a a a a a     .

Method of generating of PINC A n :

0 i m A i A n m    , ( ) 0 A A n Z  ; ( ) 1, 1; i A m i A l Z A l m l n      . Data control  1 2 1 1 1 , ,..., , , ,..., , i i i n n

A a a a a a a a

    in NRC. If > A i n N , then number à is wrong (garbled). If A i n N 

, then number А is right (ungarbled).

n М Мm            1 1

/ / 420/11 38,18 39

i n i n N N M m M m         .

Contents of the block of constants of nuvelization (BCN) concerning basis

K K   .

Resulting from the view of UC and using formula shows us that number A is wrong (Fig. 3). The represented method can be used for improving promising computer systems and their components and in the other practically important applications [5][6][7][8][9]. In particular, mathematical transformation in the system of the residue classes can be successfully adopted for optimization of calculations of cryptographic methods of data protection [10][11][12][13][14][15][24][25][26][27], also in code theories and complicated discrete signals [16][17][18][19][20][21], in authentication and steganography [22,23].

m A n A n m         ), meaning ( ) ( ) 9 A A A n Z Z  . Because of 39 9 i A N n    it

Conclusions

Thus, the method of data control in the system of residue classes is presented in the article. The procedure of forming and using of position indication of non-positional code is the base for a method of operational control of data in a residue class. Use of PINC allows to increase efficiency of the procedure of data control provided to NRC. It should be pointed out, that any non-modular operation can be implemented by set (sequence) of defined modular and non-modular operations, which are implemented by PINC. Using PINC in the method provides the possibility of exception of the most complicated positional operations from the procedure of control, diagnostic and correction of mistakes in NRC.