=Paper=
{{Paper
|id=Vol-2488/paper15
|storemode=property
|title=An Overview of Ateb-Theory Mathematical Apparatus for Data Confidentiality in Medical Computer Networks
|pdfUrl=https://ceur-ws.org/Vol-2488/paper15.pdf
|volume=Vol-2488
|authors=Ivanna Dronyuk,Olga Fedevych,Danylo Lizanets,Natalia Kryvinska
|dblpUrl=https://dblp.org/rec/conf/iddm/DronyukFLK19
}}
==An Overview of Ateb-Theory Mathematical Apparatus for Data Confidentiality in Medical Computer Networks==
https://ceur-ws.org/Vol-2488/paper15.pdf
An Overview of Ateb-Theory Mathematical Apparatus
for Data Confidentiality in Medical Computer Networks
Ivanna Dronyuk 1,3 [0000-0003-1667-2584], Olga Fedevych 1 [0000-0002-8170-3001],
Danylo Lizanets 2 [0000-0003-3675-7112], Natalia Kryvinska 3 [0000-0003-3678-9229],
1Lviv Polytechnic National University, Lviv, Ukraine
{olha.y.fedevych, ivanna.m.dronyuk}@lpnu.ua
2Wroclaw University of Science and Technology, Wroclaw, Poland
danylo.lizanets@pwr.edu.pl
3Comenius University, Bratislava, Slovakia
natalia.kryvinska@fm.uniba.sk
Abstract. In the paper, a new mathematical apparatus for confidential data
transmission in medical computer networks is proposed. It is based on the use
of Ateb algebra mathematical apparatus. It is shown, that traffic simulation
software was previously developed for Ateb-functions parameters. The pro-
posed Ateb-transform algebra and developed software traffic analyzer can be
used for further testing and investigation of medical computer networks traffic
samples, collected from other computer network topologies. The proposed ap-
proach can be used in the fields of medicine, economics, materials science, ser-
vice sciences etc., for future implementation of investigated results.
Keywords: Ateb-transform algebra, Ateb-function, medical confidentiality,
software solution, traffic flow analyzer.
1 Introduction
For the proper functioning of healthcare facilities, it is important to ensure the confi-
dentiality of data, transmitted through computer networks of healthcare facilities. In
order to solve this problem, this article proposes a method of converting signals into
the traffic of computer networks of medical institutions based on Ateb-transform,
which suppresses the usual Fourier transform [1-5,12]. This approach can be imple-
mented based on special software that is installed on all computers of the health care
facility. The use of the proposed software guarantees the confidentiality of the infor-
mation, transmitted on the network, because in case of theft information will not be
readable.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2019 IDDM Workshops.
2
2 Traffic simulation software
2.1 Previous results description
For effective network monitoring, a traffic analyzer has been developed, one of the
components of which is software for analyzing the operation of a computer network.
The purpose of the software is to provide automated collection of information from
network devices and implementation of modeling values of traffic samples. As a re-
sult of the software operation, the traffic parameters are visually displayed on the
charts and the appropriate redistribution of network load is given.
A network traffic analyzer consists of server (PHP, HTML, CSS, JS) and client (C ++
/ QT) parts. The server side deals with data analysis and storage. The client, in turn,
collects, processes network data and displays results. This software is created using C
++ and Haskell programming languages [6].
The main features of the developed traffic flow analyzer are reading samples of flow
of traffic from the nodal equipment of the computer network with the ability to record
the read results in a special file format * .pcap, analysis of previously recorded flow
of traffic. The processed traffic flow data is stored in a relational database with a reso-
lution of 1 second.
The primary purpose of a computer network traffic analyzer is to analyze samples of
that traffic, with the subsequent ability to predict the amount of traffic using Ateb-
theory. Developed software interface is shown in Figure 1. The next traffic modeling
investigation results are shown in Figures 2-9.
Fig. 1. Program window of developed traffic analyzer.
3
Fig. 2. Investigation of real and modeled traffic samples similarity based on using of Ateb-sine
with (1/3,1) parameters.
Fig. 3. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (1/3,1) parameters.
4
Fig. 4. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (1/3,1) parameters and Dirac function.
Fig. 5. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (0.01,0.1) parameters.
5
Fig. 6. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (1, 1/3) parameters.
Fig. 7. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (1, 1/7) parameters.
6
Fig. 8. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (1/3, 1) parameters.
Fig. 9. Investigation of real and modeled traffic samples approximation based on using of Ateb-
sine with (1/5, 1) parameters.
3 Review and problem statement
The introduction of the concept of T-classes became the basis for the classification of
types of properties of signals and converters in the time domain and the justification
of mathematical means of their analysis. It is also known that using a generalized shift
operator, it is possible to construct a system of signal conversion and processing
based on any orthogonal system of functions, not just a multiplication [7, 8]. This fact
is the basis for providing a new method other than harmonic analysis, signal pro-
7
cessing and transmission in wireless networks due to the ability to adapt signal pro-
cessing. The system of Ateb functions is complete and orthonormal in the functional
Hilbert space and can therefore serve as a basis [8]. To process and transmit signals, it
is necessary to have a toolkit to ensure correct addition and multiplication operations.
For this purpose, an Ateb transform algebra was constructed by the introduction of an
external convolution operation.
Here is an algebraic interpretation of the idea of T-classes with respect to Ateb func-
tions as elements of the bases of signal spaces for special communication problems.
An overview of the general state-of-the-art of the theory of generalized shift opera-
tors, hyper-complex systems, and hyper groups used to construct the Ateb-transform
algebra is presented in [9].
4 Ateb-transform algebra
Related operator for L m , n ,t : H C for Ateb-fuctions is a differential operator in a
form
d2 1
Lm,n,t () c2 ()
dt 2 , (1)
де Н- Hilbert functional space, С – set of complex numbers, c - some constant, -
the degree of nonlinearity of the operator. Ateb functions are the inherent functions of
this operator ca(m, n, t ), sa(n, m, t ) , where m, n are dependent from .
s
Now lets construct the algebra using the generalized shift operator U -such, that
U s f (t ) f (t s), where s - the value of a function argument change, interpreted as a
left shift of its function, and is therefore a family parameter {U , s R} of shift oper-
s
{Lm, n, t }
ators. Lets denote the set of operators of Ateb-transform by m, n . There-
m, n ; ,
fore, the algebra of Ateb transform can be represented as m, n where
m, n
- set of operators of Ateb-transform, - algebra`s signature m, n . The set
contains "add" and "multiply". The native operator's own functions are valid, so the
convolution operation can be defined as
( g f )(t ) g ( s ) Lsm, n, t f (t )ds.
(2)
Lets accept the specified convolution for the multiplication operation in the corre-
sponding hyper group. Adding is determined by the normal addition of features. Thus,
a linear system on the Hilbert function space is constructed, which completes within
the hyper group algebra the construction of the apparatus of spectral-temporal analy-
sis of functional spaces based on bases of Ateb functions. Parameters m, n can be
used as management and adaptation capabilities in today's communications networks.
8
4.1 Class of invariant Ateb-transforms
At this point, based on the foregoing considerations, an invariant class with respect to
the introduced transforms is constructed, which allows to close the system and show
the integrity of the constructed system.
Definition [8, 10]. The T-class is called the set of linear operators and second-order
random processes as mathematical models of systems (transducers) and signals, re-
spectively, such that the change (variability) in the time of their characteristics is de-
termined (embodied) by the operator T of generalized shift (GSO).
An analog of this definition is the fact that the term encompasses and generalizes the
notion of an invariant class when the characteristics are constant in time or formally
invariant with respect to a group of displacements on the time axis. R {Us, sR},
because Us∙Uv = Us+v, and a basis of spaces are constructing harmonics eit of fre-
quencies R, which are the own functions of the shift operator by virtue of the fact:
Us∙eit= ei(t+s) = eit∙eis, here s – parameter (shift value), and harmonic schedules
give a classic Fourier transform. In the generalizations, the concept of a hypergroup is
used instead of a group.
Lets consider D as a subspace of the space of real numbers. Then the GSO hyper-
group{Ts, sD} above the linear space of functions f: DH ( D R )are created by
operators whose eigenvalues are equal for each parameter value s GSO, by value of
own function φ(, ) a native operator of basis L {φ(, ), }, such as Lφ(,) =
∙φ(, ), that is, Tsφ(,) = φ(s,)∙φ(, ), – function index, or, in short: Ts =
φ(s,L). The basis must be orthonormal in sense
(t, ) (s, ) m(d) (t s)
, here
m() – measure on space D, the dash over the expression in this formula is a complex
conjugation, а () – Dirac delta function.
Hypergroup GSO {Ts, sD} in *-algebra (where * means involutive-convolutional
algebra) sets the external multiplication as a convolution by the formula
f * g(t) f (s)T s g(t)dt
(3)
In a linear space of functions f: DH lets define the operator of Ateb transformation
Lm,n : H C
in the form (1). Ateb functions are the inherent functions of this opera-
tor ca(m, n, t), sa(n, m, t) The system of Ateb functions at certain values of the pa-
rameters m, n is complete and ortho-normalized in the functional Hilbert space, and
therefore can serve as a basis. To process and transmit signals, it is necessary to have
the tools to ensure that the addition and multiplication operations are performed cor-
rectly. In the case of periodic Ateb-functions, the convolution is given by (3). Lets
{Lm,n }
denote the set of operators of Ateb transformation by m,n . Therefore, the
m,n ; ,
algebra of Ateb transforms can be represented as m,n where m,n – a
set of Ateb transform operators, – algebra`s signature m,n . A set contains "Addi-
tion" and "multiplication”. The multiplication above is a convolution, and the addition
9
is the usual addition of functions. Thus the totality
A m,n ;f
forms an Ateb class that
is invariant to Ateb transforms.
Lets consider F – vector space of complex-valued functions given on the set of real
numbers. It is known [8] that the linear operator L: F → F will be a generalized shift
operator if the following properties are satisfied:
1) the associativity axiom LxRy RyLx ;
2) there is an element e, the so-called neutral element, which is identical to
equation Le I , where I – identical operator.
The associative axiom can be replaced by a stronger requirement, namely the commu-
tative axiom LxLy LyLx , then the generalized shift operator is called the commuta-
tive generalized shift operator [8, 11].
If consider that the Ateb-transform operator is given by equation
Lm, n x [ A(m, n, ) iB(n, m, )]x
then it is is a commutative generalized shift op-
erator.
To show it, it is needed to show that fulfillment of the axiom of commutativity and
the existence of a neutral element.
L xL y Lm, n yLm, n x
1. m, n m, n .
Proof of this property directly follows from the commutativity of multiplication.
2. A neutral element is a function with zero values for all argument val-
L 0 0.
ues: m, n
From the shown theorem, an important consequence follows. Since the Ateb trans-
form operator is a commutative GSO and it is proved in [8, 11] that the set of GSO
creates an algebra, hence the set of Ateb transform operators also form an algebra.
{Lm,n }
Lets denote the set of operators of Ateb transform by .
So, set creates algebra under space F.
The proof follows from the previous theorem and from the statement that the set of
GSO forms an algebra [8, 10, 11].
5 Conclusions
The proposed method is new and belongs to the methods of confidential information
and data protecting based on generalized Fourier transform. The novelty is the use of
discrete Ateb-transform, which is based on Ateb-functions. In addition, a method of
protection has been developed to increase the level of data security in medical com-
puter networks in order to prevent violation of the integrity of information to ensure
the appropriate level of information transmission security.
The scope of the proposed method is quite wide; it covers different types of traffic in
various computer networks. The proposed method has been tested on traffic files. It
indicates the relevance and practical significance of the proposed method. However,
the mathematical apparatus should be improved and tested on wide samples on medi-
cal traffic data.
10
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