=Paper=
{{Paper
|id=Vol-2923/paper12
|storemode=property
|title=Prerequisites for Developing a Methodology for Estimating and Increasing Cryptographic Strength based on Many-Valued Logic Functions
|pdfUrl=https://ceur-ws.org/Vol-2923/paper12.pdf
|volume=Vol-2923
|authors=Artem Sokolov,Nadiia Kazakova,Lydia Kuzmenko,Mariia Mahomedova
|dblpUrl=https://dblp.org/rec/conf/cpits/SokolovKKM21
}}
==Prerequisites for Developing a Methodology for Estimating and Increasing Cryptographic Strength based on Many-Valued Logic Functions==
https://ceur-ws.org/Vol-2923/paper12.pdf
Prerequisites for Developing a Methodology for Estimating
and Increasing Cryptographic Strength based on Many-Valued
Logic Functions
Artem Sokolova, Nadiia Kazakovab, Lydia Kuzmenkoc, and Mariia Mahomedovac
a
Odessa National Polytechnic University, 1 Shevchenko ave., Odessa, 65044, Ukraine
b
Odessa State Environmental University, 15 Lvovskaya str., Odessa, 65016, Ukraine
c
Borys Grinchenko Kyiv University, 18/2 Bulvarno-Kudriavska str., Kyiv, 04053, Ukraine
Abstract
Symmetric cryptographic algorithms are one of the most important components of
information security systems, which determines the relevance of the task of estimation and
further increasing of their efficiency. The rapid development of cryptanalysis methods, in
particular, using the representation of structures of cryptographic algorithms based on many-
valued logic functions, and the development of quantum cryptanalysis methods, as well as
the actual absence of a comprehensive theoretical apparatus for the estimation of
cryptographic quality of many-valued logic component functions and synthesis methods for
cryptographic constructs, which are high quality in terms of many-valued logic functions,
create an objective contradiction in modern cryptography. In this paper, we represent the
prerequisites for the development of a methodology for estimation of cryptographic strength
based on many-valued logic functions, which is in turn based on the criteria and indicators of
the cryptographic quality of many-valued logic functions, as well as methods for their
calculation. We also present the results of applying the developed methodology to common
block symmetric ciphers made it possible to identify a cryptographic quality reserve that can
be obtained by improving their structure, considering the representation of their constructs
using functions of many-valued logic.
Keywords 1
Cryptography, cryptographic quality, many-valued logic function.
1. Introduction
One of the most important components of modern information security systems is the
cryptographic subsystem, which solves the problem of ensuring the integrity, confidentiality, and
authentication of transmitted and stored information. In this case, the main component of the
cryptographic subsystem is the block symmetric cipher (BSC) used for cryptographic transformation
of large amounts of data. These components are so often used in information processing and
transmission systems that the blocks corresponding to them are now implemented in processors in the
form of separate hardware modules [1]. We also note the fact that in many countries the described
cryptographic constructions are standardized and are used to protect the information in the specialized
information systems which are critical for national security. In particular, in Ukraine, the Kalyna
crypto algorithm which is described by the DSTU 7624:2014 standard [2], is used to protect the
information in military and civilian systems for processing and storing information.
These circumstances make the task of estimation and improving the cryptographic quality of these
algorithms especially urgent.
The fundamental principles on which any modern cryptographic algorithm is built are the
principles of diffusion and confusion proposed by C. Shannon [3]. However, like Shannon's theorem,
Cybersecurity Providing in Information and Telecommunication Systems, January 28, 2021, Kyiv, Ukraine
EMAIL: sokolov.a.v@opu.ua (A.1); kaz2003@ukr.net (B.2); lido4ok@gmail.com (C.3); m.mahomedova.asp@kubg.edu.ua (C.4)
ORCID: 0000-0003-0283-7229 (A.1); 0000-0003-3968-4094 (B.2); 0000-0001-7392-0324 (C.3); 0000-0001-9395-840X (C.4)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
107
the principles of diffusion and confusion give only the idea of the quality of cryptographic
constructions and the cryptographic algorithms built on their basis. Nonetheless, they provide neither
specific methods for estimation of the quality of cryptographic primitives and cryptographic
algorithms, nor methods for constructing cryptographic primitives and algorithms that would best
implement the principles of diffusion and confusion.
Since the formulation of the principles of diffusion and confusion by C. Shannon, many attempts
have been made to create a comprehensive theory for estimating the quality of cryptographic
algorithms and primitives based on them, which would be focused on the estimation of the ability of
cryptographic algorithms components and their superpositions to resist possible attacks using modern
cryptanalysis methods.
Nevertheless, the further development of cryptanalysis methods, in particular, the emergence of
cryptanalytic attacks based on many-valued logic functions [4], as well as the active development of
quantum cryptanalysis methods [5], determines the need for further improvement of methods for
estimation of the cryptographic quality of existing structures, and the development of new
cryptographic constructions, and cryptographic algorithms that meet the developed cryptographic
quality criteria.
The purpose of this paper is to research the prerequisites for creating a methodology for estimation
and increasing the cryptographic strength based on the mathematical apparatus of many-valued logic
functions.
2. Modern Methods for Cryptographic Quality Estimation
Currently, many approaches have been created to estimate the cryptographic quality, in particular,
based on the analysis of stochastic properties of the output sequences of cryptographic algorithms [6],
in which, for example, stochastic quality tests can be used [7]. There is also an approach for research
of the degree of implementation of the principles of diffusion and confusion in the cryptographic
algorithm, based on the research of its reduced copies [8], which was applied to the cryptographic
algorithms Kalyna [9], Rijndael [10] and others [11]. There is also a method for estimation of the
quality of implementation of the principles of diffusion and confusion by the cryptographic algorithm
based on the research of the number of iterations required to reach steady-state inherent to the random
substitution [12], as well as other approaches applicable for specific cryptographic algorithms.
At the same time, there is no doubt that the main cryptographic construction that determines the
quality of a cryptographic algorithm is the S-box [13].
Definition 1 [13]. A k1 k2 bit substitution block or S-box is a mapping 0,1 1 0,1 2 , that is,
k k
a mapping that uniquely maps any input k1 bit vector to an output k2 bit vector.
The structure of the S-box and, therefore, its cryptographic properties can be completely
determined by the Q-sequence of length N q k . If a given Q-sequence contains all elements of a
monotonically increasing sequence 0,1,..., N 1 , then such an S-box is called a bijective and can be
used in practical schemes of modern cryptographic algorithms. It is clear that the total cardinality of
the class of bijective S-boxes of length N is N ! .
The choice of specific S-boxes for practical use is based on their research in accordance with the
generally accepted approach, which involves the use of the mathematical apparatus of Boolean
functions. For this, the coding Q-sequence is decomposed into a set of component Boolean functions,
which can be represented using their truth tables. Further, for each of the component Boolean
functions, a set of cryptographic quality criteria is applied.
In summary, we represent in Fig. 1 a classification of the main approaches to the research of the
cryptographic quality of cryptographic algorithms.
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Figure 1: Classification of the main approaches to estimate the cryptographic quality of
cryptographic algorithms
Let us consider in detail the generally accepted approach involving the research of the
cryptographic quality of component Boolean functions. So, after representing a cryptographic
construct using Boolean functions, a set of cryptographic quality criteria is applied to them. Today,
these criteria for cryptographic quality include the following:
1. Algebraic degree of nonlinearity, which characterizes the degree of algebraic complexity of
the output of a Boolean function with respect to its input variables [14]. The algebraic degree of
nonlinearity shows how nonlinear are Boolean functions that are part of a cryptographic construction
from an algebraic point of view. When designing constructions for cryptographic algorithms,
researchers try to increase the algebraic degree of nonlinearity of their component Boolean functions.
2. Nonlinearity distance, which characterizes the degree of distancing of a Boolean function
from the set of Boolean functions that are considered linear [15]. As such a set, a set of affine
functions (codewords of the first-order Reed-Muller code) is usually used. However, to estimate the
level of nonlinearity, other constructions can also be used, which are considered linear. For example,
quadratic Boolean functions can be used as such constructions [16]. At the same time, there are two
approaches to estimate the nonlinearity of a given Boolean function: in the time domain and the
domain of Walsh-Hadamard transform coefficients.
3. Correlation relationship between the output and the input of the S-box, which is determined
by the statistical dependence of the output of the S-box on its input. For a quantitative estimation of
the level of statistical dependence, the mathematical apparatus of the matrix of correlation coefficients
[17] between the vectors of the output and the input is used. The following cryptographic quality
indicators can be applied to the correlation coefficients matrix:
a. The maximum absolute value of the elements of the matrix of correlation coefficients
between the output and input vectors.
b. The number of zero elements in the matrix of correlation coefficients. Good quality of the
cipher is the case when each of the elements of the matrix of correlation coefficients is equal to
zero, which is possible when each of the component Boolean functions of the S-box is
correlation immune [18], at least of the first order.
4. Error propagation criterion, which characterizes the ability of a cryptographic construction to
propagate minor changes in the input text or key element to the entire ciphertext [19]. There is an
error propagation criterion in the direction of a certain vector, which is fed to the input of a
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cryptographic structure as an additional influence, as well as an error propagation criterion of a given
order. The numerical determination of the correspondence of the cryptographic construction to the
error propagation criterion is carried out on the basis of the mathematical apparatus of the derivatives
of Boolean functions [14]. The error propagation criterion of the first-order is called as strict
avalanche criterion (SAC). Note, that the strict avalanche criterion is a quite strict requirement for
cryptographic constructions, therefore, in practice, the criterion of the maximum avalanche effect is
often used, which is satisfied if all weights of all derivatives of component Boolean functions in all
directions of weight 1 have a value equal to at least half the length of the cryptographic construction
[20]. The error propagation criterion, the strict avalanche criterion, and the criterion for the maximum
avalanche effect play a special role not only in the theory of analysis and synthesis of symmetric
block ciphers but also in the synthesis of cryptographically strong hash functions [21].
5. Linear redundancy of component Boolean functions. The research of component Boolean
functions [22] of some cryptographic primitives, for example, the Nyberg construction [23], shows
that they have a certain mathematical relationship with each other. This relationship weakens the level
of confusion that the researched cryptographic primitive can provide, and, accordingly, strengthens
the cryptanalyst's capabilities to describe the crypto algorithm as a whole. In [24], a new method for
determining the equivalence of component Boolean functions was proposed, which greatly simplifies
the practical task of determining the affine equivalence of component Boolean functions of
cryptographic constructions. This approach is based on a basic elementary structure definition. Thus,
to reduce the level of linear redundancy, the design of cryptographic constructions should be
performed in such a way that their component Boolean functions have a different elementary
structure.
3. Representation of Cryptographic Constructions by the Many-Valued Logic
Functions
Nevertheless, the cryptanalyst is not constrained in the chosen mathematical apparatus, with the
help of which the representation of the constructions of the cryptographic algorithm is carried out
with the subsequent attack. In addition to the mathematical apparatus of Boolean functions used in the
construction of cryptographic algorithms, the representation using 4-functions, as well as using 16-
functions, can be used for almost all modern ciphers. At the same time, the developers of
cryptographic algorithms usually do not consider these possible representations and do not research
their cryptographic quality. This circumstance determines the need to build, develop and generalize
the criteria of cryptographic quality and the practical aspects of their use for the functions of many-
valued logic.
On the other hand, quantum computers are developing dynamically, which already today allows us
to speak about the formation of post-quantum cryptographic methods that will be relevant when
quantum computers will develop and quantum attacks will be carried out with their help [5]. Note that
the key of the modern symmetric cryptographic algorithm is a pseudo-random sequence. Thus, with a
sufficiently large key length, it is required to use a brute force attack or use of any structural
vulnerability to attack the cryptographic algorithm, in contrast to the use of algorithmic attacks, for
example, using Shor's algorithm [25], which can be used to break asymmetric cryptographic
algorithms. This fact makes it especially urgent, in the conditions of post-quantum cryptography, to
research in detail and improve the structure of cryptographic algorithms, for any of their
representations, especially with the help of functions of many-valued logic.
Let us introduce the definition of a many-valued logic function we need.
Definition 1 [26]. A function of the q-valued logic of k variables is a mapping
0,1,2,..., q 1 0,1,2,..., q 1 .
k
Many-valued logic functions are more general mathematical objects than Boolean functions. So,
for value q 2 , Definition 1 is the definition of Boolean functions.
Many-valued logic is one of the experiences of expanding the boundaries of awareness and formal
description of the logical connections of the real world. With the generally accepted meaning of
binary logic, J. Łukasiewicz [27] drew attention to many-valued logic which is the way of displaying
110
different shades of information in sentences. Thus, a direction of many-valued logic arose, in which
many famous mathematicians, economists, philosophers worked, interested in improving the quality
of information transfer.
Let us consider (see Table 1) how the S-box of length N 16 practically used in modern
cryptographic algorithms can be represented not only with the help of component Boolean functions
but also with the help of many-valued logic functions.
Table 1.
S-box representation using Boolean and many-valued logic functions
Q 4 7 2 14 1 13 8 11 15 12 6 10 5 9 3 0
f 20 0 1 0 0 1 1 0 1 1 0 0 0 1 1 1 0
f 21 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0
f 22 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0
f 23 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0
f 40 0 3 2 2 1 1 0 3 3 0 2 2 1 1 3 0
f 41 1 1 0 3 0 3 2 2 3 3 1 2 1 2 0 0
At the same time, the constructions of such cryptographic algorithms as AES, Kalyna, Kuznechik,
and BelT, etc. which have an S-boxes length N 256 , can be represented using Boolean functions, 4-
functions, and 16-functions, i.e. they have 3 possible representations by many-valued logic functions.
Each of the component q-functions determines the cryptographic quality of the cryptographic
construction as a whole and, accordingly, each of them should be carefully tested.
Thus, today there is an objective contradiction between the developed methods of attacks on
cryptographic algorithms with the possible representation of their structures by functions of many-
valued logic, while simultaneously developing promising methods of attacks using quantum
computers and the actual absence of a comprehensive mathematical apparatus designed to estimate
the cryptographic quality of component functions of many-valued logic and methods for synthesis of
cryptographic structures that are high quality in terms of many-valued logic functions.
The solution to this contradiction can be obtained by developing a methodology for estimation and
increasing the cryptographic strength based on functions of many-valued logic.
The presented methodology for estimation of the cryptographic quality is based on the following
criteria for the cryptographic quality of many-valued logic functions in conjunction with the
corresponding indicators of cryptographic quality:
1. The criterion for the algebraic degree of nonlinearity of many-valued logic functions, which is
determined based on the algebraic normal form synthesized using the method [28] over a simple or
extended Galois field. At the same time, to compare the algebraic degrees of nonlinearity, the
indicator of the relative algebraic degree of nonlinearity was introduced as a percentage of a given
degree of nonlinearity of the maximum value, which allows comparing the algebraic degrees of
nonlinearity of q-functions of different lengths and for different bases q. Larger values of the relative
algebraic degree of nonlinearity indicate a higher quality of the cryptographic construction.
2. The criterion for the nonlinearity of many-valued logic functions, which is determined in
accordance with the spectral or time method [29] based on the degree of content of Vilenkin-
Chrestenson functions in the researched component function of the many-valued logic. In view of the
existence of many-valued logic bent-functions for an arbitrary given number of variables k , to
compare the nonlinearity of q-functions of different lengths and for different bases q, an indicator of
the relative nonlinearity of the q-function was introduced as a percentage of the nonlinearity of a
given q-function of the nonlinearity of bent-functions of a given length. At the same time, larger
values of the relative nonlinearity indicate a higher quality of the cryptographic structure.
3. The propagation criterion of the many-valued logic functions and the strict avalanche
criterion of the many-valued logic functions, which are determined by the research the derivatives of
many-valued logic functions in accordance with the method [30]. To assess the degree of
correspondence and comparison of q-functions of different lengths and for different bases q,
111
indicators of integral and maximum deviations from the many-valued logic functions SAC
requirements were introduced. Smaller values of the maximum and integral deviations from the SAC
requirements indicate a higher quality of the cryptographic construction.
4. The criterion for the correlation independence of the output and input vectors of
cryptographic structures [31], as well as the criterion for the independence of the output values of
many-valued logic functions from their input variables [32]. On the basis of the proposed criterion for
the independence of the output of many-valued logic functions from their input variables, indicators
of the maximum and integral deviation from the criterion for the independence of the output of many-
valued logic functions from their input are introduced, which are convenient for numerical estimation
and comparison of q-functions of different lengths and for different bases q. Smaller values of the
maximum and integral deviations from the criterion of independence of the output of many-valued
logic functions from their input indicate a higher quality of the cipher.
The described criteria and the indicators of the cryptographic quality of the many-valued logic
functions based on them provide an opportunity to estimate and compare the cryptographic structures
of various cryptographic algorithms.
At the same time, there are many practical examples when a cryptographic construction is secure
in the terms of Boolean functions and has low quality in the terms of functions of many-valued logic.
For example, the S-box S {12,7,14,9,1,4,8,3,2,6,5,11,15,10,13,0} has a nonlinearity distance value
of 4 (66.7%, which is the maximum value for S-boxes of this length) of component Boolean
functions, while the nonlinearity of 4-functions is only 3.35 (27.92%). Or the S-box
S {4,7,2,14,1,13,8,11,15,12,6,10,5,9,3,0} [30], which, being optimal from the point of view of the
strict avalanche criterion in the terms of Boolean functions, is not optimal from the point of view of
the strict avalanche criterion for component 4-functions.
Thus, the task of design of new cryptographic constructions that are cryptographically qualitative
not only from the point of view of their representation with help of Boolean functions but also from
the point of view of their representation by functions of many-valued logic seems to be urgent.
4. Evaluation of the Component Many-Valued Logic Functions Cryptographic
Quality for the Known Cryptographic Algorithms
The developed methodology for estimation and increasing cryptographic strength can be applied
both to estimate and compare the cryptographic quality of existing cryptographic algorithms and to
develop new cryptographic primitives and cryptographic algorithms. Next, we use the criteria and
indicators of cryptographic quality to estimate and compare the cryptographic properties of such well-
known cryptographic algorithms as AES [33], Kalyna [2], BelT [34], and Kuznechik [35] when
represented with help of component functions of many-valued logic. In view of the fact that the length
of the researched substitution structures of the considered cryptographic algorithms is N 256 , they
can be represented in the form of component Boolean functions, 4-functions, as well as 16-functions.
We also note that, as in the case of applying the generally accepted approach to the estimation of
the cryptographic quality based on Boolean functions, the overall cryptographic quality of a structure
is determined by its worst component q-function as the weakest component [36].
In Fig. 2 we show the results of calculating the indicators of the cryptographic quality of the many-
valued logic functions of the considered cryptographic algorithms.
The general trend is the growth of the algebraic degree of nonlinearity of existing cryptographic
algorithms when they are represented by component q-functions with larger bases q. The largest
growth is shown by BelT and Kuznechik (from 87.5% for Boolean functions to 96.67% for 16-
functions), the least growth is shown by BSC AES and Kalyna (from 87.5% for Boolean functions to
93.33% for 16-functions).
112
Figure 2: The results of calculation of the cryptographic quality indicators of the considered
cryptographic algorithms many-valued logic component functions
The behavior of the nonlinearity of component functions for different bases q depends on the
quality of the S-box used in cryptographic algorithms. At the same time, the Belarusian BSC BelT
and the Russian Kuznechik demonstrate an increase in nonlinearity when represented by many-valued
logic functions (BelT: from 86.67% for Boolean functions to 93.09% for 16-functions, Kuznechik:
from 85% for Boolean functions to 86.89% for 16-functions), while the Ukrainian BSC Kalyna
demonstrates the greatest decrease in nonlinearity when represented by many-valued logic functions
(from 86.67% for Boolean functions to 83.82% for 16-functions), which indicates less confusion that
this BSC provides in terms of many-valued logic functions.
For AES-like cryptographic algorithms, there is a general tendency towards a decrease in the
maximum deviation from the SAC with an increase in the base q of the representation of component
functions. In this case, the largest decrease in the maximum deviation from the SAC is demonstrated
by the BSC BelT (from 15.63% for Boolean functions to 4.17% for 16-functions), and the smallest
decrease is demonstrated by the BSC Kalyna (from 21.88% for Boolean functions to 6.25% for 16-
functions). There is also a general tendency for the integral deviation from the SAC to increase with
an increase in the base of the representation q. In this case, the smallest increase in the deviation from
the SAC is demonstrated by the BSC BelT (from 5.86% for Boolean functions to 14.56% for 16-
functions), and the largest increase is demonstrated by the BSC Kalyna (from 6.25% for Boolean
functions to 22.81% for 16-functions).
For AES-like cryptographic algorithms, there is also a general tendency for the maximum
deviation from the criterion of the independence of the output of component functions from the input
variables to grow with the growth of the base q of the representation of component functions. At the
same time, the lowest increase in deviation is demonstrated by the BSC BelT (from 12.5% for
113
Boolean functions to 44.9% for 16-functions), and the highest increase is demonstrated by the BSC
Kalyna (from 17.19% for Boolean functions to 57.82% for 16-functions). The integral deviation from
the criterion of independence of the output of component functions from input variables also grows.
The smallest increase in deviation is demonstrated by the BSC BelT (from 5.27% for Boolean
functions to 88.8% for 16-functions), the highest increase is demonstrated by the BSC Kalyna (from
5.88% for Boolean functions to 94.21% for 16-functions).
5. Conclusion
The research performed in this paper made it possible to draw the following conclusions:
1. At the moment, there is an objective contradiction between the developed methods of attacks
on cryptographic algorithms with the possible representation of their structures by functions of many-
valued logic, while simultaneously developing promising methods of attacks using quantum
computers and the actual absence of a comprehensive mathematical apparatus designed to research
the cryptographic quality of component functions of many-valued logic and methods for synthesizing
cryptographic structures that are high quality in terms of many-valued logic functions. The solution of
this contradiction is a prerequisite for creating a methodology for estimation and increasing
cryptographic strength based on the functions of many-valued logic.
2. A methodology for estimation and increasing cryptographic strength based on many-valued
logic functions is presented, consisting of the cryptographic quality criteria, indicators, and methods
for their calculation. The developed methodology makes it possible to estimate and compare the
cryptographic quality of existing cryptographic algorithms, as well as to develop new ones that would
be of high quality both in terms of Boolean functions and in terms of functions of many-valued logic.
3. The performed research of the common block symmetric cryptographic algorithms showed
that the representation of structures of cryptographic algorithms by functions of many-valued logic
leads to the possibility of using the proposed criteria for the cryptographic quality of functions of
many-valued logic and, accordingly, a comprehensive estimation of the cryptographic quality of these
structures. This fact leads to the need to consider the properties of many-valued logic functions when
designing cryptographic algorithms. The results of calculations of cryptographic quality indicators for
existing BSC presented in this paper confirm the possibility of further improvement of the
cryptographic algorithms considering the properties of many-valued logic functions.
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