Formalization of the Structural-functional Synthesis Problems of
Information Security Systems
Vitaly V. Gryzunov
a
and Vyacheslav G. Burlov
b
Department of Information Technology and Security Systems, Russian State Hydrometeorological University,
Voronezhskaya str. 79, Saint-Petersburg, Russian Federation
Keywords: Structural and Functional Synthesis, Synthesis of Systems With Given Properties, Emergent Properties,
Information Security System, Integrity, Information Security.
Abstract: Modern information systems tend to increase the number of nodes and users, use fog, edge and cloud
computing, spread across the territory of different states. This circumstance makes it difficult to apply the old
approaches to the synthesis of information security systems, which are based on an combination of options.
The situation can be corrected by the structural and functional synthesis of systems, during which both the
structure and functions of the system are synthesized the same time. The purpose of the article is to formalize
the tasks arising in the course of structural and functional synthesis. The article introduces the concept of a
basic pattern as a necessary attribute for structural and functional synthesis, identifies options for searching
for an approximate form of the basic laws. The corpuscular and wave properties of the system are determined.
Corpuscular properties characterize the system as an object of the material world, wave properties describe
the functions of the system. The possibility of transforming the corpuscular properties of the synthesized
system into wave and vice versa is shown. The requirements are substantiated and the axioms of the operation
of structural-functional synthesis, problems of the first, second and third kinds are formulated. Examples of
the application of the proposed formalisms for the structural-functional synthesis of a cryptoconverter are
given.
1 INTRODUCTION
Modern information systems (IS) tend to become
more complex and distributed in space-time, with the
emergence of the Internet of Things, cloud, edge and
fog computing. Enterprises are transforming to the
Industry 4.0 mode, which implies the integration of
information systems of enterprises from different
cities and even countries. Together with IS,
information security systems (ISS) are becoming
more complex. At the same time, approaches to the
synthesis of information security systems are mostly
based on an combination of possible options, which
was justified for IS at the end of the last century, but
is not quite suitable now due to the large number of
possible options.
The problem of choosing from a large number of
options is due to the very applied system synthesis
process. Currently, the synthesis of systems is
a
https://orcid.org/0000-0003-4866-217X
b
https://orcid.org/0000-0001-7603-9786
performed approximately according to the following
algorithm (GOST 34.601-90, 1992; Koller, 1976;
Muha et al., 2003):
1) to determine the purpose of the system;
2) to design the properties of the system;
3) proposing an instance of the structure, the
properties of the structure are studied (structural
synthesis is performed);
4) looking for functions that can be implemented
on the proposed structure (functional synthesis is
performed);
5) according to a certain rule, candidate elements
are selected (mainly through a survey of experts);
6) the satisfaction of the result obtained will be
checked. If the result satisfies the customer of the
system, then the system is manufactured "in metal".
At the same time, there is no guarantee that the
obtained solution is optimal according to the selected
Gryzunov, V. and Burlov, V.
Formalization of the Structural-functional Synthesis Problems of Information Security Systems.
DOI: 10.5220/0010618000003170
In Proceedings of the International Scientific and Practical Conference on Computer and Information Security (INFSEC 2021), pages 93-100
ISBN: 978-989-758-531-9; ISSN: 2184-9862
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
93

quality indicators. If the result does not satisfy the
customer of the system, then go to step 3.
7) if all options are exhausted, and the system does
not satisfy the customer, then the expected properties
of the system are adjusted and steps 3-7 are repeated.
Depending on the method used to synthesize the
system, steps 3-5 can be swapped or repeated
iteratively. In some cases, the search is directed. This
approach to synthesis is of little use for complex
systems such as modern information security
systems. They include a large number of subsystems,
which in turn are complex systems.
By a complex system we mean such a system, the
elements of which are other systems (subsystems)
(Kalinin, 2011). All systems containing subsystems
are metasystems for subsystems.
Another important disadvantage of the existing
methods for the synthesis of systems is "static".
Those. “The development of a model of any system
is carried out on the basis of a typical set of blocks
(elements), determined by the subject area and the
formulation of the task. And any typical set is actually
a made decision. As a result, the area of possible
variations in constructing the structure of the system
immediately narrows and the process of optimizing
the selected solution is difficult, which leads to the
impossibility of obtaining the required parameters at
the output of the system” (Muha et al., 2003). "Static"
makes it difficult to automatically design the
information security system and change it in real
time. This is a significant limitation for ISS, because
the environment in which ISS operates is aggressive,
focused and, most importantly, rapidly changing.
2 RESEARCH METHODS
Applying the methods of systems theory to the
synthesis of ISS, we can say that the structural-
functional synthesis of systems will allow to remove
the indicated shortcomings. When performing
structural-functional synthesis, the choice is made not
from ready-made structures with some functions, but
from the elements on the basis of which the necessary
structures are built. It is assumed that with the correct
implementation of the structural and functional
synthesis, the resulting structures immediately
provide an opportunity to implement the necessary
functions.
Summarizing the above, we can say that the
modern synthesis of systems goes in three directions
(Kun, 2003):
1. Synthesis of the structure for given functions
and algorithms of the system (structural synthesis,
functions are known).
2. Synthesis of functions, algorithms of
functioning and rules of behavior of elements of a
given hierarchical system (functional synthesis, the
structure is known).
3. Synthesis of the structure of complex systems,
including both the optimization of the functioning of
the system, and the distribution of functions among
the nodes of the system and the choice of their
composition (structural and functional synthesis, the
purpose and criteria for evaluating of the system are
known, it is necessary to find both the functions of the
system and the structure on which these functions can
be implemented).
The second direction received the greatest
elaboration. The issues of the first direction have been
worked out to a lesser extent. However, in this
direction of research, important general scientific
results were also obtained. In the third direction, as
noted in his works by one of the leading scientists
Tsvirkun A.D., there are no systemically stated
results. One of the main reasons for this, in our
opinion, is the absence of a language suitable for
structural-functional synthesis, and without language
it is impossible to develop a theory (Cvirkun, 1982).
A similar idea is expressed in (Sokolov, 2007). So,
for example, Boolean algebra became the language of
discrete mathematics and the theory of finite
automata, the language of the theory of digital signal
processing - matrices and actions with them. In
systems theory, the language of set theory and
elements of mathematical structures are used as a
basis, and then languages of other theories are used to
obtain specific results in applied fields (group theory,
differential calculus, calculus of variations, graph
theory, etc.). But the very designation of the synthesis
operation is absent. Instead of the term “synthesis”,
the term “choice” is mainly used, while it is assumed
that all the regularities are known, on the basis of
which a choice can be made from a set of alternatives,
or the algorithm by which it is necessary to make a
choice. The formation of the selection result is usually
denoted by the symbols ∪, Σ. But the operation of
combining does not say anything about the properties
of the elements, the way of combining the elements
into a system, the connection of the properties of the
elements with the properties of the entire system and
the goals of the entire system. The only those systems
can be formed by the sum of the elements, in which
there are no emergent properties.
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3 RESEARCH RESULTS
From our reasoning, it becomes clear the importance
of concepts such as the goal and property of the
system. Let us consider them in more detail, then we
will formulate the problem of structural-functional
synthesis of systems and show what properties the
operations should have, allowing to carry out
structural-functional synthesis.
3.1 Basic Definitions of
Structural-functional Synthesis
Many works are devoted to the concept of property,
which are mainly philosophical. Having studied such
works, one can understand the meaning of the
concept, but it is difficult to use it in formalized
operations. Mathematicians and scientists of natural
sciences, as a rule, study some specific properties:
properties of functions, groups, matter, light, etc., and
not the concept itself, as such.
In (Encyclopedic Dictionary, 2009), the property
is defined as follows.
PROPERTY - a feature inherent in an object and
allowing it to be included in a particular class of
objects. Distinguish between essential (substantial)
properties of an object and insignificant, accidental -
accidents.
PROPERTY, a philosophical category that
expresses the relationship of a given thing to other
things with which it interacts. Property is often
viewed as an external expression of quality.
We will refer to the substantial properties as
properties, without which the object will not be able
to realize its purpose (to achieve the goal of its
existence). It follows from the definition that a
property is manifested only in interaction and is its
characteristic (expresses a relation). In mathematics,
the rule that characterizes the interaction is called a
mapping. Let us formulate the definition of a property
in set-theoretic form. Property - is a mapping of a set
X (an object, the owner of a property) into a set Y (an
object with which interaction is organized):
𝑋→𝑌
It follows from the definition that the appearance
of a new object with which interaction is organized
can lead to the appearance of new properties in the
original object. This is true. For example, any object
in the dark has a black color, and the color scale
appears only in the presence of light (interaction of
the object and light). A computer without an energy
source has no performance, however, when energy is
supplied, productivity appears (the interaction of a
computer and energy).
In nature, as a rule, all studied subjects are
systems, therefore we will further understand a
subject as a system. Any system consists of elements.
We classify the properties of the system on the
following grounds and describe them in the proposed
notation:
 by the way of creating:
- properties of the system, which are reduced to
the properties of the elements of the system according
to a certain rule. Such properties are specified by
mapping the elements of one (original) set to the
elements of the same set or a set obtained by
combining elements from the original set: 𝑅:𝑋

⟶
𝑋

∪𝐵

𝑋


, where 𝑋

is the set of properties of
elements, 𝐵

𝐴

is a boolean, given on the set A. For
example, mass (formed by the sum of the masses of
elements), volume (formed by transforming the
volumes of elements), the probability of no-failure
operation (formed by transforming the probabilities
of no-failure operation of elements), a binary function
(0 and 1 are fed into the input, 0 and 1) etc.
- system properties that are not reducible to
element properties are emergence property: 𝑋

,
𝑋

∩𝑋

=∅. For such properties, the mapping of
elements of one set to elements of another set is
specified: 𝑅:𝑋

∪𝐵

𝑋


→𝑋

. For example, the
maximum speed of a vehicle (engine power, drag
coefficient, mass, etc. is assigned a new element -
speed). The maximum flight altitude (energy
capacity, aerodynamic characteristics, engine power,
etc., the new element is assigned the distance to the
Earth's surface), etc.
 by the way of presentation (let's draw an
analogy with the wave-particle concept):
- corpuscular 𝑋

- characterizing the system and
its elements, as an object of the material world:
reliability, color, mass, etc.
- following the laws of formal logic, we must
divide the properties into corpuscular and non-
corpuscular, among the latter to single out wave.
However, at the moment, no other ways of
representing an object, except for corpuscular and
wave, are known, therefore we will assume that all
non-corpuscular properties are wave 𝑋

and
characterize the functions of the system.The set of all
properties of the 𝑋

system can be represented as
follows: 𝑋

∪𝑋

, or 𝑋

=𝑋

∪𝑋

.
Let us call the mapping R, which allows us to
obtain the properties of the entire system from the
properties of the elements, the basic law of the
system's functioning. Any other laws are not basic for
the system. Basic laws are described in terms of
theories from which the system is considered. For
example, for an unmanned aerial vehicle, the basic
Formalization of the Structural-functional Synthesis Problems of Information Security Systems
95

laws can be Newton's laws (as for a kinematic
system), Kirchhoff's and Ohm's laws (as for an
electrical system), laws in pattern recognition theory
(as for an intelligent system), economic laws (as for
an object with a value ) etc. Please note that we have
not said anything about the mapping type. It can be
anything: function, functional, clear, fuzzy, etc.
In the theory of systems, dynamical systems are
considered, as a rule, given in the terminal form:
𝑆=

𝑇,𝑄,𝑋,𝑌,𝜑,𝜓

(1)
where
T is the set of points in time at which the system
operates.
Q is the set of input situations, determined by the
set of system inputs. The only tool to influence the
properties of the system.
Y is the set of output situations, determined by the
set of outputs of the system, we will call it simple
properties, i.e. properties that can be measured
directly.
X - a set (space) of states of the system - the
motion of a dynamic system - constitutes internal
properties.
ψ: T × Q × X⟶Y is the output mapping. A
transformation according to which simple properties
can be derived from intrinsic properties. If there is a
transformation that makes it possible to obtain
internal ones from simple properties, the system is an
observable according to Kalman.
φ: T × Q × X⟶X is a transition mapping. A
transformation that directly affects the intrinsic
properties. If with its help it is possible to achieve any
state from the set of admissible ones, then the system
is controllable.
Let us call such representation (1) wave, i.e.
representation of the system through its functions. If
the system is given in the form of a graph, a reliability
scheme, etc., then such a representation will be called
corpuscular, i.e. representation of the system through
its structure. Applying the above to the operations of
synthesis, let us say that as a result of structural
synthesis we obtain a system in a corpuscular
representation, as a result of functional synthesis - in
a wave representation. Obviously, the result of
structural-functional synthesis should be a wave-
particle representation of the system, or such a
representation of the system, from which it is easy to
pass to the corpuscular or wave one.
The concept of the goal and quality of the system
helps to single out the substantial ones from all the
properties of the system.
The goal (Lopatnikov, 2003) in economic
cybernetics, systems analysis is the desired state of
system outputs (final state) as a result of a controlled
process of its development. It is set by the goal
determination unit, which is included in the control
subsystem. The states of the system (as well as its
trajectories) are evaluated from the point of view of
their conformity or inconsistency with goal. The
mathematical expression (model) of such an
evaluation is the objective function or the quality
criterion of the system (in the case of system
optimization, the optimality criterion).
In other words, the goal of the system's
functioning is specified by forming in the sets
𝑇,𝑄,𝑋,𝑌 the values of interest to the creator of the
system: 𝑡
∗
∈𝑇,𝑞
∗
∈𝑄,𝑥
∗
∈𝑋,𝑦
∗
∈𝑌 . In the
general case, the values of interest are sets, and the
goal itself is supplemented by criteria P, according to
which the best is selected from the set of possible
movements of the system leading to the goal.
Quality is the degree of conformity of an object to
its purpose (Petuhov, 1989).
Mathematically, the presence of quality in a
system can be written as ∃𝑡
∗
∈𝑇,𝑞
∗
∈𝑄,𝑥
∗
∈
𝑋,𝑦
∗
∈𝑌, i.e. the properties of a quality system
always allow the system to reach its goal.
If several variants of the system and its movement
are possible, leading the system to the goal, you need
to choose the best option according to the criteria set
when formulating the goal. In order to assess the
possible options for the system, consider such a
property as integrity.
To determine the integrity, we will use the
description given in (Hoode and Machol, 1962).
“Every large-scale system has a certain integrity.
Although the system may not be tightly controlled
from one central location, all parts of the system serve
some common purpose; in a sense, they all contribute
to the development of a certain set of optimal outputs
from a given set of inputs, and the optimality is
assessed according to a certain criterion of
efficiency". Using the above, we define the integrity
of the system ( 𝑅
∝
) as a property showing how
consistent the elements of the system are with each
other, how they help the system to achieve the goal of
its functioning. In other words, are the functions ϕ and
ψ optimal according to the established criteria P, i.e.
how efficiently (fast, cheap, accurate, etc.) they bring
the system to the goal 𝑡
∗
∈𝑇,𝑞
∗
∈𝑄,𝑥
∗
∈𝑋,𝑦
∗
∈𝑌.
We define 𝑅
∝
=0 for complete inconsistency of
system elements and 𝑅
∝
=1 for complete
consistency.
The considered concepts are enough to formalize
the problem of structural and functional synthesis.
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3.2 Formalization of the Problem of
Structural-functional Synthesis
Structural synthesis Σ

∝
is an operation that results in
the formation of the corpuscular properties of the
system.
Functional synthesis Σ

∝
is an operation that
results in the formation of the wave properties of the
system.
Structural-functional synthesis Σ
∝
=Σ

∝

↔Σ

∝

is an operation that results in the formation of both
corpuscular and wave properties of the system.
Includes operations of structural and functional
synthesis, interconnected through basic laws.
By the number of known initial data, the problem
of structural and functional synthesis has varying
complexity.
3.2.1 The Third Kind of
Structural-functional Synthesis
Problems
The initial data are maximal. The classical
optimization problem, i.e. the problem of choosing
from a variety of alternatives.
Given
𝑠∈𝑆

- a set of elements available for synthesis
with X_ev properties.
𝑋

∗
=𝑋

,𝑋

 - required system properties.
𝑡
∗
∈𝑇,𝑞
∗
∈𝑄,𝑥
∗
∈𝑋,𝑦
∗
∈𝑌 are the goals of the
system.
P - criteria for choosing the best version of the
system.
R - basic laws that allow obtaining the properties
of the system from the properties of individual
elements.
It is required to find
𝑆=Σ
∝

𝑋

,𝑅,𝑆


:𝑥

≥𝑥

∗
,∀𝑥

∈𝑋

,𝑅
∝
→1 is
an optimal system with 𝑋

properties, synthesized
from available elements with 𝑋

properties.
The solution to the problem of the third kind is
currently the most studied. A typical example is the
synthesis of systems, the functions of which can be
written down analytically. In these cases, the form of
the function, as a rule, immediately defines the
structure: finite automata (Boolean function), control
systems (transfer function), etc. The converse is also
true, i.e. a ready-made structure defines a function
that is implemented by the structure.
An example of the formulation of the task of
synthesizing an information security system in
(Hoode and Machol, 1962) "from the set of possible
options for information security with given external
system relations for control and interaction in the
structure of the organizational and technical system
(OTS), it is required to determine the set of
admissible options for information security that
ensure the specified efficiency of using the OTS in a
conflict" .
The problem of the third kind for the synthesis of
an information security system using graph theory is
solved in (Mistrov, 2009). In the article (Kustov,
Jakovlev and Stankevich, 2017), the author reduces
the problem of synthesizing the information security
system to the optimal justification of quantitative and
qualitative requirements for the information security
system at an acceptable cost. In the study
(Tatarnikova, 2013), on the basis of the terminal
model of the communication management system, the
actions of a social engineer violating information
security are synthesized, communication tools and
methods are linked, the requirements for the structure
and feedback of communication are substantiated, the
necessary communication algorithms are selected
depending on the observed reaction of the
communication object.
3.2.2 The Second Kind of
Structural-functional Synthesis
Problems
Given
𝑠∈𝑆

- a set of elements available for synthesis
with X_ev properties.
𝑋

∗
=𝑋

,𝑋

 - required system properties.
𝑡
∗
∈𝑇,𝑞
∗
∈𝑄,𝑥
∗
∈𝑋,𝑦
∗
∈𝑌 are the goals of the
system.
P - criteria for choosing the best version of the
system.
It is required to find
R - basic laws that allow obtaining the properties
of the system from the properties of individual
elements.
𝑆=Σ
∝

𝑋

,𝑅,𝑆


:𝑋

≥𝑋

∗
,𝑅
∝
→1 is an
optimal system with 𝑋

, properties, synthesized from
available elements with 𝑋

properties.
In the process of solving a problem of the second
kind, the most difficult thing is the search for basic
laws, which is carried out, as a rule, in various fields
of science. For example, for a spacecraft, the basic
laws will be the laws of mechanics, and electrical
engineering, and discrete mathematics, etc. After the
basic laws have been established, the problem of the
second kind is simplified and becomes a problem of
the third kind.
The problem can be partially solved if the basic
law is sought approximately. In this case, we are
Formalization of the Structural-functional Synthesis Problems of Information Security Systems
97

talking about the use of neural networks, genetic
algorithms, adaptation, etc. In (Gryzunov and
Bondarenko, 2018), a DDoS attack detection system
is synthesized. Different traffic properties are fed to
the method input: the sequence of packet arrival, time
intervals between packets, etc. The basic law by
which the ISS is synthesized is not known, is sought
in an approximate form using Kohonen maps. In the
study (Chistohodova and Sidorov, 2017), in the form
of basic laws, interscheme properties are generated,
which are generated with the help of an intermediary
in a semi-automatic mode.
The paper (Palopoli, Terracina and Ursino, 2000)
considers the structural-parametric synthesis of an
information security system from elements certified
by FSTEC. The synthesis is carried out on the basis
of a genetic algorithm in stages: the choice of the
structure, the selection of parameters. The ISS
requirements are set by the user in the form of the
required IS security class.
3.2.3 The First Kind of
Structural-functional Synthesis
Problems
The initial data is minimal.
Given
𝑡
∗
∈𝑇,𝑞
∗
∈𝑄,𝑥
∗
∈𝑋,𝑦
∗
∈𝑌 are the goals of the
system.
P - criteria for choosing the best version of the
system.
It is required to find
𝑠∈𝑆

- a set of elements required for synthesis
with 𝑋

properties.
𝑋

∗
=𝑋

,𝑋

 - required system properties.
R - basic laws that allow obtaining the properties
of the system from the properties of individual
elements.
𝑆=Σ
∝

𝑋

,𝑅,𝑆


:𝑋

≥𝑋

∗
,𝑅
∝
→1 is an
optimal system with 𝑋

properties, synthesized from
available elements with 𝑋

properties.
The problem of the first kind is the most difficult
and demanded one. In this case, the customer of the
system describes the purpose of the system (goals),
formulates the criteria for choosing the best version
of the system (performance criteria). According to the
requirements and restrictions put forward, the
properties of the system are formulated, the search for
basic laws and the selection of elements made up the
system are made.
Thus, in the process of solving the problem of
structural-functional synthesis, it is necessary to solve
all the existing problems of scientific research
(Kalinin, 2011) (modeling, analysis of properties,
observation, choice). Let us consider the operation of
structural-functional synthesis itself in more detail.
One of the steps to solving the problem of the first
kind is presented in (Zhukov, 2016). The authors
propose the rules by which the hierarchy of
information protection efficiency indicators is
formed: from indicators of individual elements of the
system to the indicator of the efficiency of the system
as a whole.
Attempts to synthesize an intrusion detection
system by solving a problem of the first kind are
presented in (Dzhogan, Kurilo and Shimon, 2011).
Researchers in (Gryzunov, 2006) synthesize a safety
management system for a technosphere object.
3.3 Operation of Structural-functional
Synthesis
The operation of structural-functional synthesis must
meet the following requirements:
 to provide the ability to dock the corpuscular
and wave representations of the system;
 to be scalable, applicable to all levels of the
system hierarchy;
 to have variable arity, since at the beginning of
the operation it is not known how many
elements the finished system will contain.
The range of the operation definition is a set of
properties of the 𝑋

system. The operation binds the
properties of the system to each other, therefore the
range of values of the operation is also 𝑋

.
The operation is defined over a set, which means
it forms an algebraic structure. Let us introduce the
basic axioms of the operation (Burlov, Andreev and
Gomazov, 2018).
Existence of a neutral element
In the system, you can always find such a property
that does not in any way affect the final result we
expect. Such a property will be a neutral element
∃𝑒 ∈ 𝑋

,∀𝑎∈𝑋

:Σ
∝

𝑒,𝑎

=Σ
∝

𝑎,𝑒

=Σ
∝

𝑎

.
The presence of a reverse element
Analysis of existing systems shows that death
processes always coexist with the processes of
reproduction, that for any body there is an antibody,
etc. This allows us to make the assumption that there
is always an inverse element in the system. ∃𝑎

∈
𝑋

,∀𝑎∈𝑋

:Σ
∝

𝑎

,𝑎

=Σ
∝

𝑎,𝑎


=Σ
∝

𝑒

.
Associativity
In general, Σ
∝
is not associative, therefore, groups
cannot be formed on its basis. Algebraic structures
that are not groups, as well as operations with variable
arity, are currently the least studied, probably this is
another reason for the poor study of issues of
structural and functional synthesis. However, in those
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cases when it is possible to impart associative
properties to the synthesis operation, and to form
groups or even Abelian groups, structural-functional
synthesis is relatively simple.
Let's consider an example of the application of the
introduced concept.
4 RESULTS DISCUSSION
Formalization of the problem of structural-functional
synthesis of the first kind
Suppose we need to synthesize a cryptographic
transformer (CT) with a final performance Ω .
Objective Ω∈

Ω

,Ω

,Ω


, performance evaluation
criterion 𝑃:Ω → 𝑚𝑎𝑥.
Formalization of the problem of structural and
functional synthesis of the second kind
Let CTs with capacities Ω

,Ω

,…,Ω

be
available to us. This problem can be solved using 3
CTs with performance Ω

,Ω

,Ω

, which can be
switched on parallel or serial.
An additional property of the i-th CT is (works /
does not work)

L

=1,L


=0

.
X

∗
=X

=

Ω

,Ω

,Ω


,

L

,L





,

L

,L





,

L

,L






Formalization of the problem of structural and
functional synthesis of the third kind
The number of operating CPs will be called the
configuration of the final CT 𝑄=L

+2L

+4L

.
When the elements are connected in parallel, the
actual performance does not exceed the total 𝑅

:Ω≤
∑
Ω



. With serial 𝑅

:Ω≤min
;




Ω

. =

𝑅

,𝑅


.
The formalization of the problem of the structural-
functional type of the third is completed.
The solution to the developed problem is a parallel
and, possibly, serial and mixed connection of the CTs.
Given our criterion for evaluating efficiency, we
choose a parallel connection. The solution can be
written in the form of a logical-dynamic operator that
simultaneously sets the corpuscular (fig. 1) and wave
representation (fig.2) of the final cryptographic
transformer.
Σ
∝
=Σ
∝


↔Σ
∝

=L

Ω

+L

Ω

+L

Ω

Figure 1: The corpuscular representation of CT.
Figure 2: The wave representation of CT (CT phase space).
K is the number of tasks solved by the final CT.
For the practical implementation of structural-
functional synthesis, appropriate methods are
required.
5 CONCLUSIONS
The development of structural-functional synthesis is
impossible without an appropriate language.
Depending on the completeness of the initial data,
three kinds of problems of structural-functional
synthesis are possible. The problem of choosing from
a variety of alternatives, which is being solved today
in the process of systems synthesis, is part of the
problem of structural-functional synthesis. The basic
law required for synthesis can be found
approximately using neural networks, genetic
algorithms, methods of adaptive control theory, etc.
The operation of structural-functional synthesis
forms an algebraic structure that is not a group, does
not have associativity, has variable arity, neutral and
inverse elements.
Further, for the practical application of the
concept, it is necessary to develop:
Formalization of the Structural-functional Synthesis Problems of Information Security Systems
99

 methods of formalizing the target purpose of
the functioning of the system, searching for
criteria for evaluating the system;
 methods of basic laws searching;
 methods that make it possible to reasonably
deduce the requirements for the elements from
the basic laws (the required number of
elements, the main functions of the elements,
etc.).
The concept proposed in the article can be used
not only for the synthesis of information security
systems and technical systems, but also for any other
dynamic systems, for example, chemical elements,
troops, state structure, etc., for this it is necessary to
formulate the basic laws and develop appropriate
methods.
ACKNOWLEDGEMENTS
The reported study was funded by Russian Ministry
of Science (information security), project № 08/2020.
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