=Paper=
{{Paper
|id=Vol-2353/paper19
|storemode=property
|title=Method of Data Control in the Residue Classes
|pdfUrl=https://ceur-ws.org/Vol-2353/paper19.pdf
|volume=Vol-2353
|authors=Viсtor Krasnobayev,Alexandr Kuznetsov,Anna Kononchenko,Tetiana Kuznetsova
|dblpUrl=https://dblp.org/rec/conf/cmis/KrasnobayevKKK19
}}
==Method of Data Control in the Residue Classes==
https://ceur-ws.org/Vol-2353/paper19.pdf
Method of data control in the residue classes
Victor Krasnobayev [0000-0001-5192-9918], Alexandr Kuznetsov [0000-0003-2331-6326],
Anna Kononchenko [0000-0002-8101-6500], Tetiana Kuznetsova [0000-0001-6154-7139]
V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine
v.a.krasnobaev@gmail.com, kuznetsov@karazin.ua,
akononpro@gmail.com, kuznetsova.tatiana17@gmail.com
Abstract. Methods of data control in the residue classes are considered in the
article. The main advantage of non-positional notation in the residue classes
lays in the possibility of an organization of the process of quick implementation
of modular arithmetic operations of addition, subtraction and multiplication.
The base of the method is the procedure of generating and using the positional
indication of non-positional code. That allows increasing efficiency of the pro-
cedure of data control granted by the residue classes.
Keywords. Methods of data control, the system of the residue classes, the posi-
tional indications of the non-positional code, computer systems and compo-
nents.
1 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 so-
called non-modular (position) operations except the above listed arithmetic opera-
tions. The following operations belong to these:
Arithmetic and algebraic congruence of numbers and its absolute values;
Sign of number definition;
Definition of existing of overflow of bit grid of computer system (CS);
Rounding of value of result of operation;
Computing the absolute value;
Division and multiplication of fraction
Translation of data from code of NRC to positional notation and back;
Expanding of original NRC (it is an informational process in which familiar re-
mainders {ai } , that correspond to basis {mi } , define value of remainders with the
same code structure by other additional basis);
Control, diagnostic and correction mistakes of NRC data, etc.
Generally, all positional operations come down to the procedure of definition of an
index of j numerical jmi , j 1 mi interval of entering (detecting) of number
А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 . It is appropriate to use so-called positional indi-
cations 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 yn 1 );
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
А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 . 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 non-
modular 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 ]
n
interval, where M mi ) by used (chosen, developed, generated) PINC;
i 1
Simplicity of generating PINC for given code structure of data
А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 ;
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 opera-
tions.
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.
2 Method of generating of positional indication of non-
positional code structure of data in NRC
Looking upon method of generating of PINC (Fig. 1).
The indication n A forms proceeding from representation of initial code structure
А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 , which is represented as NRC with ba-
sis {mi } , i 1, n 1 , in the way of creating of so-called uniserial code (UC).
In standard form, UC
K N( nA ) Z N( A)1 Z N( A)1 ... Z1( A) Z 0( A)
represents sequence of binary bits, which contains ones and a zero, index of which is
at n A place (the count is performed from right to left, from digit bit Z 0( A) to digit bit
Z N( A)1 ). Indication of PINC n A defines an index of j numerical interval
jmi , j 1 mi of detecting of number А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 .
Mathematically PINC n A represents a positive integer, which points out the location
of zero binary bit in the record of UC K N A
(n )
Z ( A)
nA
0 .
The procedure of generating UC K N( nA ) lies in the following. A constant
KH m( A) a1' ,..., ai' 1 , ai , ai' 1 ,..., an' 1
i
defines in the block of constants of nuvelization (BCN). It is performed by the defini-
tion of remainder ai of number А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 for chosen inte-
ger mi of NRC.
Further, using the chosen constant of nuvelization KH m( A) , we displace number A
i
on the left edge of the interval jmi , j 1 mi by implementing operation
Am A KH m
i
( A)
i
' ' '
a1 , a2 , ..., ai 1 , ai , ai 1 , ..., an 1 a1 , a2 , ..., ai 1 , ai , ai 1 ,..., an 1
' '
a1 , a2 , ..., ai 1 , 0, ai 1 , ..., an 1 .
(1) (1) (1) (1) (1)
Choice informational {m i } , i 1, n and control mk mn 1
1. ( mi mi 1 ) basis of residue classes for representing data
A (a1 , a2 ,..., ai 1 , ai , ai 1 ,..., an , an 1 ) , GCD ( mi , m j ) 1 , i j .
Choice of basis mi {m i } , ( j 1, n 1) , by which defines an index of j
2.
numerical interval jmi , j 1 mi of detecting number А.
Defining the constant of reducing to zero polynomial
3. KH m( A) (a1' , a2' ,..., ai' 1 , ai , ai' 1 ,..., an' , an' 1 )
i
by value of remainder ai (by module mi ) of number A.
Defining the value
Ami A KH m( A) a1 , a2 ,..., ai 1 , ai , ai 1 ,..., an , an 1
4. i
a1' , a2' ,..., ai' 1 , ai , ai' 1 ,..., an , an 1 a , a ,..., a , 0, a ,..., a , a .
(1)
1
(1)
2
(1)
i 1
(1)
i 1
(1)
n
(1)
n 1
Defining UC
K N( nA ) Z N( A)1 Z N( A)2 ... Z0( A) , K N( nA ) Z N( A) 1 Z N( A) 2 ... Z0( A) .
i i i
n 1 n
N mK , N i ]M / mi [ , M mi . Ami K A mi Z K( A) .
A
K 1 i 1
K i
Am 0 mi Z ( A) ,
A i 1 m Z (0A) ,
5. mi i 1
... ( A)
Ami ( N 2) mi Z N 2 ,
( A)
Ami ( N 1) mi Z N 1.
A 0 m Z ( A) ,
mi i
(0A)
Ami 1 mi Z1 ,
... ( A)
Ami ( Ni 2) mi Z Ni 2 ,
Am ( Ni 1) mi Z A) .
(
i Ni 1
Defining PINC n A of number A, which means numerical value of n A for
6 which Z K( A) Z n( A) 0 , which means Ami n A mi 0 .
A A
Herewith Z l( A) 1, ( Ami l mi 0; l n A ) .
Fig. 1. The method of generating PINC in NRC
It is obvious, that number A is aliquot to value of module mi of NRC.
It is known, that an indication of the correctness of number A in NRC defines by
it's entering or not entering in its numerical informational interval [0, M ) . If number
A is out of the interval ( A M ) , A is considered as garbling (wrong). In this case,
PINC n A has to define the fact of entering or not entering initial number A in the in-
terval [0, M ) . We have to implement operation
Ami K A mi Z K( A) (1)
A
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
K A mi K A 0, N 1 ,
n 1
where N mK :
K 1
K i
Am 0 mi Z ( A) ,
0
i ( A)
Ami 1 mi Z1( A),
Am 2 mi Z ,
i ... 2 (2)
( A)
Ami ( N 2) mi Z(NA) 2 ,
Ami ( N 1) mi Z N 1.
In this case, UC represents in the view
K N( nA ) Z N( A)1 Z N( A)2 ... Z1( A) Z0( A) (3)
The only value n A from (1) exists in total (2) analytical rations. For n A there is
Z K( A) Z n( A) 0 ( K A n A ) ,
A A
which means
Ami n A mi 0 .
The rest of values (2) equal
Zl( A) 1 ( Ami l mi 0; l n A ) .
In the general case, the amount of N binary bits in the record UC K N( nA ) equal to
value
n 1
N mK .
K 1
K i
It should be pointed out, that there is no necessity to have the whole sequence of
values Z K( A) from (3) to define only the fact of the garbling of number ( A M ) . It is
A
(n )
enough to have UC K N A with length only Ni ]M / mi [ of binary bits (where
i
value ]M / mi [ describes the lesser integer of number M / mi ).
In this case values of variables of numerical intervals jmi , j 1 mi , which are
out of informational interval 0, M , make no matter for establishing the fact of cor-
rectness control of number A.
3 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 А а1 ,..., аi –1 , аi , аi 1 ,..., an , an 1 , which is repre-
sented in residue classes, developed (defined) PINC n A by generating UC
i
K N( nA ) Z N( A) 1 Z N( A) 2 ... Z1( A) Z 0( A)
i i
in view of sequence of Ni binary bits. Choosing of basis mi of NRC is performed by
special-purpose in accordance with defined measures. The constant of nuvelization
KH m( A) a1' , a2' ,..., ai ,..., an' , an' 1
i
is selected depending on a result of the value of remainder ai of number A. Further
implementation operation is carried out:
Ami A KH m( A) a1 , a2 ,..., ai ,..., an , an 1 a1' , a2' ,..., ai ,..., an' , an' 1
i
a1(1) , a2(1) ,..., 0,..., an(1) , an(1)1 .
Using Ni constants K A mi K A 0, Ni 1 simultaneously the operations of sub-
tractions Ami K A mi are carried out, in the result of which appears the values of
(n )
binary bits Z K( A) , so the UC K N A forms.
A i
The values of PINC n A defined from the equation Ami n A mi 0 .
If n A Ni , number A is considered as a wrong. In the opposite case ( n A Ni )
number A is correct. In common view, method of data control is represented on Fig. 2.
Algorithm of defining UC
i
K N( nA ) Z N( A) 1 Z N( A) 2 ... Z1( A) Z 0( A)
i i
of number
A a1 , a2 ,..., ai 1 , ai , ai 1 ,..., an , an 1 .
Method of generating of PINC n A :
Ami n A mi 0 , Z n( A) 0 ; Z l( A) 1, Ami l mi 1; l n A .
A
Data control A a1 , a2 ,..., ai 1 , ai , ai 1 ,..., an , an 1 in NRC.
If n A > Ni , then number à is wrong (garbled).
If n A Ni , then number А is right (ungarbled).
Fig. 2. Method of data control in NRC
Examine now the examples of implementing of method of control for specific
NRC, which is given with basis m1 3, m2 4, m3 5, m4 7 and
mk mn 1 m5 11 . The NRC provides the data handling in single-byte (1=1)
bit grid of CS. Herewith
4
M mi 420 , М 0 Мmn 1 4620;
i 1
N i N n 1 M / mi M / mn 1 420 / 11 38,18 39 .
Contents of the block of constants of nuvelization (BCN) concerning basis
mk mn 1 11 are given in Table 1.
Example 1. Perform control of data that represented in
view A (01, 00, 000, 010, 0001) .
Constant of nuvelization
KH m( A) (01, 01, 001, 001, 0001)
n 1
is selected by values of remainders аК an 1 a5 0001 of number A in BCN CS
(Table 1). Then we define value
( A)
Table 1. Constants KH m of nulevization BCN
n 1
Constant of nuvelization
Remainder
m1 3 m2 4 m3 5 m4 7 mk m5 11
ak an 1
a1 a2 a3 a4 a5
0000 00 00 000 000 0000
0001 01 01 001 001 0001
0010 10 10 010 010 0010
0011 00 11 011 011 0011
0100 01 00 100 100 0100
0101 10 01 000 101 0101
0110 00 10 001 110 0110
0111 01 11 010 000 0111
1000 10 00 011 001 1000
1001 00 01 100 010 1001
1010 01 10 000 011 1010
Amn 1 A KH m( A) (00,11,100, 001, 0000) .
n 1
By implementation of ratio (2) we form UC
(n ) (9)
K N A K39 {11...110111111111} .
i
Resulting from the view of UC and using formula
Amn 1 n A mn 1 0 ,
we define that n A 9
( Amn 1 n A mn 1 99 9 11 0 ),
meaning Z n( A) Z9( A) .
A
Because of Ni 39 nA 9 it means that there is no mistake in data.
Check: А 100 M 420 (number A is right).
Example 2. Perform control of data A (00,10, 000, 010,1010) .
The constant
KH m( A) (01,10, 000, 011,1010)
n 1
is selected by value of a5 1010 in BCN (Table 1).
We deduce that
Amn 1 A KH m( A) (10, 00, 000,110, 0000) .
n 1
Because
Amn 1 n A mn 1 440 44 11 0
(n ) (40)
then UC has view K N A K39 {11...11...11} and n A 40 .
i
Because of Ni 39 n A 40 it means that there is a mistake in data.
Check: А 450 M 420 (number A is wrong).
Example 3. Perform control of data А 01,11, 010, 000,1001 . .
The constant
KH m( A) (00, 01,100, 010,1001)
n 1
is selected by value of a5 1001 in BCN (Table 1). We deduce that
Amn 1 A KH m( A) (01,10, 011,101, 0000) .
n 1
Because
Amn 1 n A mn 1 418 38 11 0
then UC has view
(n ) (38)
K N A K39 {011...11...11}
i
and n A 38 .
Because n A 38 Ni 39 of it means that number A is right (ungarbled).
Though the check А 427 M 420 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 [5-9]. In par-
ticular, mathematical transformation in the system of the residue classes can be suc-
cessfully adopted for optimization of calculations of cryptographic methods of data
protection [10-15, 24-27], also in code theories and complicated discrete signals [16-
21], in authentication and steganography [22, 23].
4 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.
Fig. 3. Example of implementing operation of data control in NRC for mn 1 m5 11
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 imple-
mented 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.
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